Traffic circle warning system and method

ABSTRACT

A traffic circle warning system and method employ a controller. The controller is configured to determine whether a traffic circle exists along a current travel path of the host vehicle based on remote vehicle information representing a travel condition of at least one remote vehicle. The controller is further configured to, upon determining that the traffic circle exists, evaluate a travel condition of the host vehicle relative to the traffic circle and the travel condition of the remote vehicle to determine whether to control a warning system onboard the host vehicle to issue a warning.

CROSS-REFERENCE TO RELATED APPLICATIONS

Related subject matter is disclosed in U.S. patent application Ser. No.15/477,827, entitled “Traffic Circle Identification System and Method,”,filed concurrently herewith. The entirety of the “Detailed Descriptionof the Embodiments,” and the entirety of all of the Figures, of U.S.patent application Ser. No. 15/477,827 entitled “Traffic CircleIdentification System and Method,” is incorporated by reference herein.

BACKGROUND OF THE INVENTION Field of the Invention

The present invention generally relates to a traffic circle warningsystem and method. More specifically, the present invention relates toan on-board vehicle system and method for evaluating travel conditionsof a host vehicle and a remote vehicle relative to a traffic circle todetermine whether to control a warning system onboard the host vehicleto issue a warning.

Background Information

Vehicles having a navigation system typically acquire and store road mapdata that the navigation system uses to generate a map display. A mapdisplay typically includes images representing the roads within adesignated area of the vehicle, as well as other images such aslandmarks, fueling station locations, restaurants, weather data, trafficinformation and so on.

Traffic circles are becoming more common, especially to avoid the use oftraffic signals in highly traveled areas. As drivers understand, trafficcircles are different to navigate than typical intersections. Therefore,it can be beneficial for a driver to be informed of the presence of anupcoming traffic circle in advance, and whether there should be anyconcern for other vehicles that are in or approaching the trafficcircle. Map data is currently the most common way of detecting thepresence of a traffic circle in a vehicle's path.

SUMMARY OF THE INVENTION

Although map data can be used to identify traffic circles, it ispossible that a vehicle may be unable to acquire accurate map data incertain locations. For example, map data may not take into accountrecently constructed traffic circles if the map data is out of date.Therefore, a need exists for an improved traffic circle warning systemfor identifying a traffic circle, especially along a current travel pathof a host vehicle, and determining whether to issue a warning to thedriver of the host vehicle based on the location and movement of anyother vehicle in or near the traffic circle.

In accordance with one aspect of the present invention, a traffic circlewarning system and method are provided which employ a controller. Thecontroller is configured to determine whether a traffic circle existsalong a current travel path of the host vehicle based on remote vehicleinformation representing a travel condition of at least one remotevehicle. The controller is further configured to, upon determining thatthe traffic circle exists, evaluate a travel condition of the hostvehicle relative to the traffic circle and the travel condition of theremote vehicle to determine whether to control a warning system onboardthe host vehicle to issue a warning.

These and other objects, features, aspects and advantages of the presentinvention will become apparent to those skilled in the art from thefollowing detailed description, which, taken in conjunction with theannexed drawings, discloses a preferred embodiment of the presentinvention.

BRIEF DESCRIPTION OF THE DRAWINGS

Referring now to the attached drawings which form a part of thisoriginal disclosure:

FIG. 1 is a schematic diagram illustrating an example of a host vehicleequipped with a traffic circle warning system according to embodimentsdisclosed herein, in relation to remote vehicles and components of aglobal positioning system (GPS) and a communication system;

FIG. 2 is a block diagram of exemplary components of the host vehicleequipped with a traffic circle warning system according to embodimentsdisclosed herein;

FIG. 3 is a diagrammatic view illustrating an example of a condition inwhich a remote vehicle is in the traffic circle and is about to crossthe path of the host vehicle which is entering the traffic circle;

FIG. 4 is a diagrammatic view illustrating an example of a condition inwhich a remote vehicle is in the traffic circle and is ahead of the hostvehicle after crossing the path of the host vehicle;

FIG. 5 is a diagrammatic view illustrating an example of a condition inwhich a remote vehicle is in the traffic circle on the opposite side ofthe traffic circle from the host vehicle and diverging from the hostvehicle;

FIG. 6 is a diagrammatic view illustrating an example of a condition inwhich a remote vehicle is in the traffic circle on the opposite side ofthe traffic circle from the host vehicle and converging with the hostvehicle;

FIG. 7 is a diagrammatic view illustrating an example of a condition inwhich the host vehicle is in the traffic circle and the remote vehicleis approaching the traffic circle and is about to cross the path of thehost vehicle;

FIG. 8 is a diagrammatic view illustrating an example of a condition inwhich the host vehicle is in the traffic circle and is ahead of theremote vehicle after crossing the path of the remote vehicle;

FIG. 9 is a diagrammatic view illustrating an example of a condition inwhich the host vehicle is in the traffic circle on the opposite side ofthe traffic circle from the remote vehicle and diverging from the remotevehicle;

FIG. 10 is a diagrammatic view illustrating an example of a condition inwhich the host vehicle is in the traffic circle on the opposite side ofthe traffic circle 40 from the remote vehicle and converging with theremote vehicle;

FIG. 11 is a flowchart illustrating an example of operations performedby the traffic circle warning system to determine whether a warningshould be issued due to the location of the host vehicle and at leastone remote vehicle with respect to the traffic circle;

FIGS. 12-19 are graphical representations of a location of the hostvehicle with respect to a remote vehicle as used in calculationsperformed by the traffic circle warning system during the operation ofthe flowchart of FIG. 11;

FIGS. 20-43 are graphical representations of heading angles of the hostvehicle and the remote vehicle in relation to each other as used incalculations performed by the traffic circle warning system during theoperation of the flowchart of FIG. 11;

FIG. 44 is a diagrammatic representation of an example of thecalculations performed by the traffic circle warning system during theoperation of the flowchart of FIG. 11 to determine whether a warningshould be issued; and

FIG. 45 is a flowchart illustrating an example of operations performedby the traffic circle warning system to issue a warning.

DETAILED DESCRIPTION OF THE EMBODIMENTS

Selected embodiments of the present invention will now be explained withreference to the drawings. It will be apparent to those skilled in theart from this disclosure that the following descriptions of theembodiments of the present invention are provided for illustration onlyand not for the purpose of limiting the invention as defined by theappended claims and their equivalents.

Referring initially to FIG. 1, a two-way wireless communications networkis illustrated that includes vehicle to vehicle communication andvehicle to base station communication. In FIG. 1, a host vehicle (HV) 10is illustrated that is equipped with a traffic circle warning system 12according to a disclosed embodiment, and two remote vehicles (RV) 14that also includes the traffic circle warning system 12. As discussedherein, the host vehicle 10 can also be referred to as a subject vehicle(SV). The remote vehicle 14 can also be referred to as a target orthreat vehicle (TV). While the host vehicle (HV) 10 and the remotevehicles 14 are illustrated as having the same traffic circle warningsystem 12, it will be apparent from this disclosure that each of theremote vehicles 14 can include another type of two-way communicationsystem that is capable of communicating remote vehicle informationrepresenting a travel condition of the remote vehicle 14 to the hostvehicle 10. The remote vehicle information can include, for example,information representing the location (e.g., GPS location), speed,acceleration and heading of the remote vehicle 14 at each of a pluralityof locations of the remote vehicle 14, information representing arespective turning radius of the remote vehicle 14 at each of theplurality of locations of the remote vehicle 14, turn signal activationat the remote vehicle 14 at each of the plurality of locations, and anyother type of information suitable for representing a travel path of theremote vehicle 14. Likewise, the host vehicle 10 can also exchange hostvehicle information with each of the remote vehicles 14. This hostvehicle information can include, for example, information representingthe location (e.g., GPS location), speed, acceleration and heading ofthe host vehicle 10 at each of a plurality of locations of the hostvehicle 10, information representing a respective turning radius of thehost vehicle 10 at each of the plurality of locations of the hostvehicle 10, turn signal activation at the host vehicle 10 at each of theplurality of locations, and any other type of information suitable forrepresenting a travel path of the host vehicle 10. The host vehicle 10and the remote vehicles 14 can exchange this type of host vehicleinformation and remote vehicle information with each other several timesper second, or at any suitable time intervals.

The traffic circle warning system 12 of the host vehicle 10 and theremote vehicle 14 communicates with the two-way wireless communicationsnetwork. As seen in FIG. 1, for example, the two-way wirelesscommunications network can include one or more global positioningsatellites 16 (only one shown), and one or more roadside (terrestrial)units 18 (only one shown), and a base station or external server 20. Theglobal positioning satellites 16 and the roadside units 18 send andreceive signals to and from the traffic circle warning system 12 of thehost vehicle 10 and the remote vehicles 14. The base station 20 sendsand receives signals to and from the traffic circle warning system 12 ofthe host vehicle 10 and the remote vehicles 14 via a network of theroadside units 18, or any other suitable two-way wireless communicationsnetwork.

As shown in more detail in FIG. 2, the traffic circle warning system 12includes an application controller 22 that can be referred to simply asa controller 22. The controller 22 preferably includes a microcomputerwith a control program that controls the components of the trafficcircle warning system 12 as discussed below. The controller 22 includesother conventional components such as an input interface circuit, anoutput interface circuit, and storage devices such as a ROM (Read OnlyMemory) device and a RAM (Random Access Memory) device. Themicrocomputer of the controller 22 is at least programmed to control thetraffic circle warning system 12 in accordance with the flow chart ofFIG. 8 as discussed below. It will be apparent to those skilled in theart from this disclosure that the precise structure and algorithms forthe controller 22 can be any combination of hardware and software thatwill carry out the functions of the present invention. Furthermore, thecontroller 22 can communicate with the other components of the trafficcircle warning system 12 discussed herein via, for example a controllerarea network (CAN) bus or in any other suitable manner as understood inthe art.

As shown in more detail in FIG. 2, the traffic circle warning system 12can further include a wireless communication system 24, a globalpositioning system (GPS) 26, a storage device 28, a plurality ofin-vehicle sensors 30 and a human-machine interface unit 32. Thewireless communication system 24 can include, for example, atransmitter, a receiver, a transceiver, and any other suitable type ofequipment as understood in the art. The human-machine interface unit 32includes a screen display 32A, an audio speaker 32B and various manualinput controls 32C that are operatively coupled to the controller 22.The screen display 32A and the audio speaker 32B are examples ofinterior warning devices of a warning system that are used to alert adriver. Of course, it will be apparent to those skilled in the art fromthis disclosure that interior warning devices include anyone of or acombination of visual, audio and/or tactile warnings as understood inthe art that can be perceived inside the host vehicle 10. The hostvehicle 10 also includes a pair of front headlights 34 and rear brakelights 36, which constitutes examples of exterior warning devices of thetraffic circle warning system 12. These components can communicate witheach other and, in particular, with the controller 22 in any suitablemanner, such as wirelessly or via a vehicle bus 38.

The wireless communications system 24 can include an omni-directionalantenna and a multi-directional antenna, as well as communicationinterface circuitry that connects and exchanges information with aplurality of the remote vehicles 14 that are similarly equipped, as wellas with the roadside units 20 through at least a portion of the wirelesscommunications network within the broadcast range of the host vehicle10. For example, the wireless communications system 24 can be configuredand arranged to conduct direct two way communications between the hostand remote vehicles 10 and 14 (vehicle-to-vehicle communications) andthe roadside units 18 (roadside-to-vehicle communications). Moreover,the wireless communications system 24 can be configured to periodicallybroadcast a signal in the broadcast area. The wireless communicationsystem 24 can be any suitable type of two-way communication device thatis capable of communicating with the remote vehicles 14 and the two-waywireless communications network. In this example, the wirelesscommunication system 24 can include or be coupled to a dedicated shortrange communications (DSRC) antenna to receive, for example, 5.9 GHzDSRC signals from the two-way wireless communications network. TheseDSRC signals can include basic safety messages (BSM) defined by currentindustry recognized standards that include information which, undercertain circumstances, can be analyzed to warn drivers of a potentialproblem situation or threat in time for the driver of the host vehicle10 to take appropriate action to avoid the situation. For instance, theDSRC signals can also include information pertaining to weatherconditions, adverse driving conditions and so on. In the disclosedembodiments, a BSM includes information in accordance with SAE StandardJ2735 as can be appreciated by one skilled in the art. Also, thewireless communication system 24 and the GPS 26 can be configured as adual frequency DSRC and GPS devices as understood in the art.

The GPS 26 can be a conventional global positioning system that isconfigured and arranged to receive global positioning information of thehost vehicle 10 in a conventional manner. Basically, the globalpositioning system 26 receives GPS signals from the global positioningsatellite 16 at regular intervals (e.g. one second) to detect thepresent position of the host vehicle 10. The GPS 26 has an accuracy inaccordance with industry standards and thus, can indicate the actualvehicle position of the host vehicle 10 within a few meters or less(e.g., 10 meters less). The data representing the present position ofthe host vehicle 10 is provided to the controller 22 for processing asdiscussed herein. For example, the controller 22 can include or becoupled to navigation system components that are configured and arrangedto process the GPS information in a conventional manner as understood inthe art.

The storage device 28 can store the remote vehicle information asdiscussed above. The storage device 28 can also store road map data, aswell as other data that can be associated with the road map data such asvarious landmark data, fueling station locations, restaurants, weatherdata, traffic information and so on. Furthermore, the storage device 28can store other types of data, such as data pertaining tovehicle-related parameters and vehicle conditions. For example, thevehicle-related parameters can include predetermined data indicatingrelationships between vehicle speed, vehicle acceleration, yaw, steeringangle, etc. when a vehicle is preparing to make a turn. In this event,the storage device 28 can further store data pertaining to vehicleconditions, which can represent a determined vehicle condition of avehicle of interest, such as the host vehicle 10, a remote vehicle 14,or both. This determined vehicle condition can represent, for example, avehicle speed and acceleration that is determined for the vehicle ofinterest at a moment in time. Accordingly, the embodiments disclosedherein can evaluate whether the vehicle condition lies within the areaof interest, as represented by the vehicle-related parameters, todetermine, for example, whether the vehicle of interest is preparing tomake a turn. The storage device 28 can include, for example, alarge-capacity storage medium such as a CD-ROM (Compact Disk-Read OnlyMemory) or IC (Integrated Circuit) card. The storage device 28 permits aread-out operation of reading out data held in the large-capacitystorage medium in response to an instruction from the controller 22 to,for example, acquire the map information and/or the vehicle conditioninformation as needed or desired for use in representing the location ofthe host vehicle 10, the remote vehicle 14 and other locationinformation and/or vehicle condition information as discussed herein forroute guiding, map display, turning indication, and so on as understoodin the art. For instance, the map information can include at least roadlinks indicating connecting states of nodes, locations of branch points(road nodes), names of roads branching from the branch points, placenames of the branch destinations, and so on. The information in thestorage device 28 can also be updated by the controller 22 or in anysuitable manner as discussed herein and as understood in the art.

The in-vehicle sensors 30 are configured to monitor various devices,mechanisms and systems within the host vehicle 10 and provideinformation relating to the status of those devices, mechanisms andsystems to the controller 22. For example, the in-vehicle sensors 30 canbe connected to a traction control system, a windshield wiper motor orwiper motor controller, a headlight controller, a steering system, aspeedometer, a braking system and so on as understood in the art.

Examples of operations performed by the traffic circle warning system 12will now be discussed with reference to FIGS. 3 to 45. As can beappreciated from the following description, because the host vehicle 10and the remote vehicles 14 are equipped with vehicle to vehiclecommunication technology as discussed above, the host vehicle 10 can usethe remote vehicle information received from other similarly equippedremote vehicles 14 to determine the presence and size of a trafficcircle without need for map data, which can provide a significant costsavings. Also, in view of pending NHTSA regulations that would requirevehicle to vehicle communication technology in new vehicles in thefuture, the traffic circle warning system 12 according to the disclosedembodiments can significantly enhance the functionality of crash warningsystems that leverage information received via vehicle to vehiclecommunication from other vehicles to either suppress warnings that arenot necessary, or issue warnings under circumstances that othersensor-based systems could not detect. For instance, by using GPSposition and heading information received from remote vehicles 14, thetraffic circle warning system 12 according to the disclosed embodimentsprovides an accurate identification of the presence and size of anapproaching traffic circle. This information can be used to suppressunnecessary warnings that could otherwise be a nuisance. The trafficcircle warning system 12 also provides a very rapid detection ofwrong-way driving of a remote vehicle 14, as well as the host vehicle10, that may be travelling in the wrong direction around the trafficcircle. The traffic circle warning system 12 can also be beneficial withregard to compliance with Federal Motor Vehicle Safety Standards (FMVSS)and New Car Assessment Program (NCAP) requirements.

As can be appreciated from FIGS. 3 through 10, unlike a traditionalintersection where threat of contact with another vehicle can occur fromall directions, only two scenarios exist in a traffic circle where acontact may occur. One condition is when the host vehicle 10 isapproaching the traffic circle 40, and another is when the host vehicle10 is in the traffic circle 40. When the host vehicle 10 is approachingthe traffic circle 10, the threat of contact with a remote vehicle 14only exists when a remote vehicle 14 in the traffic circle 40 is aboutthe cross the path of the host vehicle 10 as shown, for example, in FIG.3. That is, FIG. 3 is a diagrammatic view illustrating an example of acondition in which a remote vehicle 14 is in the traffic circle and isabout to cross the path of the host vehicle 10 which is entering thetraffic circle 40.

However, under the other instances shown in FIGS. 4 through 6, thelikelihood of the host vehicle 10 and the remote vehicle 14 contactingeach other is extremely remote. For example, FIG. 4 is a diagrammaticview illustrating an example of a condition in which a remote vehicle 14is in the traffic circle 40 and is ahead of the host vehicle 10 aftercrossing the path of the host vehicle 10. FIG. 5 is a diagrammatic viewillustrating an example of a condition in which a remote vehicle 14 isin the traffic circle 40 on the opposite side of the traffic circle 40from the host vehicle 10 and diverging from the host vehicle 10. FIG. 6is a diagrammatic view illustrating an example of a condition in which aremote vehicle 14 is in the traffic circle 40 on the opposite side ofthe traffic circle 40 from the host vehicle 10 and converging with thehost vehicle 10.

However, FIG. 7 is a diagrammatic view illustrating an example of acondition in which the host vehicle 10 is in the traffic circle 40 andthe remote vehicle 14 is approaching the traffic circle 40 and is aboutto cross the path of the host vehicle 10. Thus, a threat of contactbetween the host vehicle 10 and the remote vehicle 14 exists. In allother instances shown in FIGS. 8 through 10, the likelihood of contactbetween the host vehicle 10 and the remote vehicle 14 is extremelyremote. For example, in FIG. 8 is a diagrammatic view illustrating anexample of a condition in which the host vehicle 10 is in the trafficcircle 40 and is ahead of the remote vehicle 14 after crossing the pathof the remote vehicle 14. FIG. 9 is a diagrammatic view illustrating anexample of a condition in which the host vehicle 10 is in the trafficcircle 40 on the opposite side of the traffic circle 40 from the remotevehicle 14 and diverging from the remote vehicle 14. FIG. 10 is adiagrammatic view illustrating an example of a condition in which thehost vehicle 10 is in the traffic circle 40 on the opposite side of thetraffic circle 40 from the remote vehicle 14 and converging with theremote vehicle 14.

FIG. 11 is a flowchart illustrating an example of operations performedby the traffic circle warning system 12 to determine whether a warningshould be issued due to the location of the host vehicle 10 and at leastone remote vehicle 14 with respect to the traffic circle 40. In Step100, the traffic circle warning system 12 receives remote vehicleinformation from at least one remote vehicle 14. As discussed above, theremote vehicle information can include, for example, informationrepresenting the location (e.g., GPS location), speed, acceleration andheading of the remote vehicle 14 at each of a plurality of locations ofthe remote vehicle 14, information representing a respective turningradius of the remote vehicle 14 at each of the plurality of locations ofthe remote vehicle 14, turn signal activation at the remote vehicle 14at each of the plurality of locations, and any other type of informationsuitable for representing a travel path of the remote vehicle 14. Asalso discussed above, the host vehicle 10 can exchange host vehicleinformation with the remote vehicle 14. This host vehicle informationcan include, for example, information representing the location (e.g.,GPS location), speed, acceleration and heading of the host vehicle 10 ateach of a plurality of locations of the host vehicle 10, informationrepresenting a respective turning radius of the host vehicle 10 at eachof the plurality of locations of the host vehicle 10, turn signalactivation at the host vehicle 10 at each of the plurality of locations,and any other type of information suitable for representing a travelpath of the host vehicle 10. The host vehicle 10 and the remote vehicles14 can exchange this type of host vehicle information and remote vehicleinformation with each other several times per second, or at any suitabletime intervals.

In Step 102, the traffic circle warning system 12 can analyze the remotevehicle information to determine whether the traffic circle 40 exists,the diameter of the traffic circle 40, and the location of any remotevehicle 14 with respect to the host vehicle 10 and the traffic circle40, without using or relying upon map data. For example, the trafficcircle warning system 12 onboard the host vehicle 10 stores GPS positionheading and speed information in the remote vehicle information receivedfrom the remote vehicle 14. The software being run by the controller 22can include, for example, a software application onboard the hostvehicle 12 to use this remote vehicle information to calculate thelocation of the remote vehicle 14 in relation to the host vehicle 10 andthe traffic circle 40 as will now be described. For purposes of thedescription below, the host vehicle 10 is represented by “HV” and theremote vehicle 14 is represented by “RV” in the following equations,tables and graphs.

The controller 22 can define a series of mathematical expressions thatprovide specific information regarding the longitudinal, lateral,elevation and heading of a remote vehicle 14 relative to the hostvehicle 10. These equations are used to determine the position of theremote vehicle 14 relative to the host vehicle 10 in order to determineif a threat condition exists.

The following exemplary equation is used to determine the longitudinaland lateral position of a remote vehicle 14 relative to the host vehicle10. Using the coordinates of North, South, East and West with the hostvehicle 10 being at the center purposes of these examples and equations,the processing performed by the controller 22 can divide the areasurrounding the host vehicle 10 into quadrants Q1, Q2, Q3 and Q4 as willnow be described. By performing these operations, the controller 22 iseffectively identifying sections of the traffic circle 40 sincedepending upon the location of the host vehicle 10, at least some of thequadrants Q1, Q2, Q3 and Q4 can overlap with at least a portion of thetraffic circle 40.

FIGS. 12 and 13 illustrate a condition in which the remote vehicle 14 isto the Northeast of the host vehicle 10, and thus is in quadrant Q1.

Q1: Remote Vehicle 14 is to the Northeast of the Host Vehicle 10

$Q_{1} = {{\frac{1}{4}\left\lbrack {\frac{\phi_{RV} - \phi_{HV} - \sigma}{{{\phi_{RV} - \phi_{HV}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{\theta_{RV} - \theta_{HV} + \sigma}{{{\theta_{RV} - \theta_{HV}}} + \sigma} + 1} \right\rbrack}$

If the remote vehicle 14 is northeast of the host vehicle 10 as shown inFIGS. 12 and 13, both latitude and longitude for the remote vehicle 14is greater than the latitude and longitude for the host vehicle 10.Under these conditions, the expression for Q₁ above will equal 1otherwise it will equal 0.

Longitudinal Position (XW)

The remote vehicle 14 is ahead (XW=00) of the host vehicle 10 if:0≤δ_(HV) <A ₁, or A ₂≤δ_(HV)<2π

where:

A₁=β₁+π/2−φ₁

A₄=β₁+3π/2+φ₁

φ₁ is a threshold value that defines the angular range in which theremote vehicle 14 is defined to be adjacent to the host vehicle 10.

This region β₁ calculated by the following equation is identified by thevertical cross-hatching

in FIG. 12.

$\beta_{1} = {{\pi\left\lbrack {\frac{\theta_{HV} - \theta_{RV} - \sigma}{{{\theta_{HV} - \theta_{RV}}} + \sigma} + 1} \right\rbrack} - {{\cos^{- 1}\left( \frac{\left( {\phi_{RV} - \phi_{HV}} \right)}{\sqrt{{\left( {\theta_{RV} - \theta_{HV}} \right)^{2}\cos^{2}\phi_{HV}} + \left( {\phi_{RV} - \phi_{HV}} \right)^{2}}} \right)}\left\lbrack \frac{\theta_{HV} - \theta_{RV} - \sigma}{{{\theta_{HV} - \theta_{RV}}} + \sigma} \right\rbrack}}$

and these conditions can be defined in one mathematical expression as:

$P_{Q_{1}} = {{{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - 0 + \sigma}{{{\delta_{HV} - 0}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{1} - \delta_{HV} - \sigma}{{{A_{1} - \delta_{HV}}} + \sigma} + 1} \right\rbrack} + {{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{4} + \sigma}{{{\delta_{HV} - A_{4}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{{2\;\pi} - \delta_{HV} - \sigma}{{{{2\;\pi} - \delta_{HV}}} + \sigma} + 1} \right\rbrack}}$

The remote vehicle 14 is adjacent (XW=01) to the host vehicle 10 if:A ₁≤δ_(HV) <A ₂ or A ₃≤δ_(HV) <A ₄

where:

A₁=β₁+π/2−φ₁

A₂=β₁+π/2+φ₁

A₃=β₁+3π/2−φ₁

A₄=β₁+3π/2+φ₁

These two specific angular ranges are identified by the checkeredcross-hatching

as shown in FIG. 12, which is also the interfaces between the areaidentified by the vertical cross-hatching

as discussed above, and the area identified by the slantedcross-hatching

as discussed below. These conditions can be defined in one mathematicalexpression as:

$A_{Q_{1}} = {{{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{1} + \sigma}{{{\delta_{HV} - A_{1}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{2} - \delta_{HV} - \sigma}{{{A_{2} - \delta_{HV}}} + \sigma} + 1} \right\rbrack} + {{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{3} + \sigma}{{{\delta_{HV} - A_{3}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{4} - \delta_{HV} - \sigma}{{{A_{4} - \delta_{HV}}} + \sigma} + 1} \right\rbrack}}$

The remote vehicle 14 is behind (XW=10) the host vehicle 10 if:A ₂≤δ_(HV) <A ₃

where:

A₂=β₁+π/2+φ₁

A₃=β₁+3π/2−φ₁

and this region is identified as by the slanted cross-hatching

in FIG. 12. These conditions can be defined in one mathematicalexpression as:

$B_{Q_{1}} = {{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{2} + \sigma}{{{\delta_{HV} - A_{2}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{3} - \delta_{HV} - \sigma}{{{A_{3} - \delta_{HV}}} + \sigma} + 1} \right\rbrack}$

Lateral Position (VU):

The remote vehicle 14 is in lane (VU=00) with the host vehicle 10 if:A ₅≤δ_(HV) <A ₆ or A ₇≤δ_(HV) <A ₈

where:

A₅=β₁−φ₂

A₆=β₁+φ₂

A₇=β₁+π−φ₂

A₃=β₁+π+φ₂

φ₂ is a threshold value that defines the angular range in which theremote vehicle 14 is defined to be in the same lane with the hostvehicle 10.

These two specific angular ranges are identified by the verticalcross-hatching

in FIG. 13, which is also the interfaces between the area identified bythe checkered cross-hatching

as shown in FIG. 13 and the area identified by the horizontalcross-hatching

in FIG. 13, as discussed below.

These conditions can be defined in one mathematical expression as:

$I_{Q_{1}} = {{{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{5} + \sigma}{{{\delta_{HV} - A_{5}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{6} - \delta_{HV} - \sigma}{{{A_{6} - \delta_{HV}}} + \sigma} + 1} \right\rbrack} + {{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{7} + \sigma}{{{\delta_{HV} - A_{8}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{8} - \delta_{HV} - \sigma}{{{A_{8} - \delta_{HV}}} + \sigma} + 1} \right\rbrack}}$

and the remote vehicle 14 is to the left (VU=01) of the host vehicle 10if:A ₆≤δ_(HV) <A ₇

where:

A₆=β₁+φ₂

A₇=β₁+π−φ₂

This region is identified by the horizontal

cross-hatching in FIG. 13. These conditions can be defined in onemathematical expression as:

$L_{Q_{1}} = {{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{6} + \sigma}{{{\delta_{HV} - A_{6}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{7} - \delta_{HV} - \sigma}{{{A_{7} - \delta_{HV}}} + \sigma} + 1} \right\rbrack}$and the remote vehicle 14 is to the right (VU=10) of the host vehicle 10if:0≤δ_(HV) <A ₅ or A ₈≤δ_(HV)<2π

where:

A₅=β₁−φ₂

A₈=β₁+α+φ₂

This region is identified by the checkered cross-hatching

as shown in FIG. 13. These conditions can be defined in one mathematicalexpression as:

$R_{Q_{1}} = {{{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - 0 + \sigma}{{{\delta_{HV} - 0}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{5} - \delta_{HV} - \sigma}{{{A_{5} - \delta_{HV}}} + \sigma} + 1} \right\rbrack} + {{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{8} + \sigma}{{{\delta_{HV} - A_{8}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{{2\pi} - \delta_{HV} - \sigma}{{{{2\pi} - \delta_{HV}}} + \sigma} + 1} \right\rbrack}}$

The expressions are then consolidated in the following matrix for thecase when the remote vehicle 14 is to the northeast of the host vehicle10.

Q₁ Lateral Position RV in lane (I_(Q) ₁ ) RV Left (L_(Q) ₁ ) RV Right(R_(Q) ₁ ) Unused Longitudinal RV Ahead (P_(Q) ₁ ) Q₁ × P_(Q) ₁ × I_(Q)₁ Q₁ × P_(Q) ₁ × L_(Q) ₁ Q₁ × P_(Q) ₁ × R_(Q) ₁ 0 Position RV Adjacent(A_(Q) ₁ ) Q₁ × A_(Q) ₁ × I_(Q) ₁ Q₁ × A_(Q) ₁ × L_(Q) ₁ Q₁ × A_(Q) ₁ ×R_(Q) ₁ 0 RV Behind (B_(Q) ₁ ) Q₁ × B_(Q) ₁ × I_(Q) ₁ Q₁ × B_(Q) ₁ ×L_(Q) ₁ Q₁ × B_(Q) ₁ × R_(Q) ₁ 0 Unused 0 0 0 0

FIGS. 14 and 15 illustrate a condition in which the remote vehicle 14 isto the Northwest of the host vehicle 10, and is in quadrant Q2.

Q2: Remote Vehicle 14 is to the Northwest of the Host Vehicle 10

$Q_{2} = {{\frac{1}{4}\left\lbrack {\frac{\phi_{RV} - \phi_{HV} + \sigma}{{{\phi_{RV} - \phi_{HV}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{\theta_{HV} - \theta_{RV} - \sigma}{{{\theta_{HV} - \theta_{RV}}} + \sigma} + 1} \right\rbrack}$

If the remote vehicle 14 is northwest of the host vehicle as shown inFIGS. 14 and 15, the latitude for the remote vehicle 14 is greater thanthe latitude of the host vehicle 10, but the longitude for the remotevehicle 14 is less than the longitude for the host vehicle 10. Underthese conditions, the expression for Q2 above will equal 1 otherwise itwill equal 0.

Longitudinal Position (XW)

The remove vehicle 14 is ahead (XW=00) of the host vehicle 10 if:0≤δ_(HV) <A ₉ or A ₁₂≤δ_(HV)<2π

where:

A₉=β₁−3π/2−φ₁

A₁₂=β₁−π/2+φ₁

φ₁ is a threshold value that defines the angular range in which the RVis defined to be adjacent to the HV.

This region β₁ calculated by the following equation is identified by thevertical cross-hatching

in FIG. 14.

$\beta_{1} = {{\pi\left\lbrack {\frac{\theta_{HV} - \theta_{RV} - \sigma}{{{\theta_{HV} - \theta_{RV}}} + \sigma} + 1} \right\rbrack} - {{\cos^{- 1}\left( \frac{\left( {\phi_{RV} - \phi_{HV}} \right)}{\sqrt{{\left( {\theta_{RV} - \theta_{HV}} \right)^{2}\cos^{2}\phi_{HV}} + \left( {\phi_{RV} - \phi_{HV}} \right)^{2}}} \right)}\left\lbrack \frac{\theta_{HV} - \theta_{RV} - \sigma}{{{\theta_{HV} - \theta_{RV}}} + \sigma} \right\rbrack}}$

These conditions can be defined in one mathematical expression as:

$P_{Q_{1}} = {{{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - 0 + \sigma}{{{\delta_{HV} - 0}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{9} - \delta_{HV} - \sigma}{{{A_{9} - \delta_{HV}}} + \sigma} + 1} \right\rbrack} + {{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{12} + \sigma}{{{\delta_{HV} - A_{12}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{{2\;\pi} - \delta_{HV} - \sigma}{{{{2\;\pi} - \delta_{HV}}} + \sigma} + 1} \right\rbrack}}$

and the remote vehicle 14 is adjacent (XW=01) to the host vehicle 10 if:A ₉≤δ_(HV) <A ₁₀ or A ₁₁≤δ_(HV) <A ₁₂

where:

A₉=β₁−3π/2−φ₁

A₁₀=β₁−3π/2+φ₁

A₁₁=β₁−π/2−φ₁

A₁₂=β₁−π/2+φ₁

These two specific angular ranges are identified by the checkeredcross-hatching

as shown in FIG. 14 as the interfaces between the area identified by thevertical cross-hatching

and the area identified by the slanted cross-hatching

in FIG. 14. These conditions can be defined in one mathematicalexpression as:

$A_{Q_{1}} = {{{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{9} + \sigma}{{{\delta_{HV} - A_{9}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{10} - \delta_{HV} - \sigma}{{{A_{10} - \delta_{HV}}} + \sigma} + 1} \right\rbrack} + {{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{11} + \sigma}{{{\delta_{HV} - A_{11}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{12} - \delta_{HV} - \sigma}{{{A_{12} - \delta_{HV}}} + \sigma} + 1} \right\rbrack}}$

and the remote vehicle 14 is behind (XW=10) the host vehicle 12 if:A ₁₀≤δ_(HV) <A ₁₁

where:

A₁₀=β₁−3π/2+φ₁

A₁₁=β₁−π/2−φ₁

This region is identified by the slanted cross-hatching

in FIG. 14. These conditions can be defined in one mathematicalexpression as:

$B_{Q_{2}} = {{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{10} + \sigma}{{{\delta_{HV} - A_{10}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{11} - \delta_{HV} - \sigma}{{{A_{11} - \delta_{HV}}} + \sigma} + 1} \right\rbrack}$

Lateral Position (VU)

The remote vehicle 14 is in lane (VU=00) with the host vehicle 10 if:A ₁₃≤δ_(HV) <A ₁₄ or A ₁₅≤δ_(HV) <A ₁₆

where:

A₁₃=β₁−πφ₂

A₁₄=β₁−π+φ₂

A₁₅=β₁−φ₂

A₁₆=β₁+φ₂

φ₂ is a threshold value that defines the angular range in which theremote vehicle 14 is defined to be in the same lane with the hostvehicle 10.

These two specific angular ranges are identified by the verticalcross-hatching

as the interfaces between the area identified by the checkeredcross-hatching

and the area identified by the horizontal cross-hatching

in FIG. 15.

These conditions can be defined in one mathematical expression as:

$I_{Q_{2}} = {{{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{13} + \sigma}{{{\delta_{HV} - A_{13}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{14} - \delta_{HV} - \sigma}{{{A_{14} - \delta_{HV}}} + \sigma} + 1} \right\rbrack} + {{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{15} + \sigma}{{{\delta_{HV} - A_{15}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{16} - \delta_{HV} - \sigma}{{{A_{16} - \delta_{HV}}} + \sigma} + 1} \right\rbrack}}$

and the remote vehicle 14 is to the left (VU=01) of the host vehicle 10if:0≤δ_(HV) <A ₁₃ or A ₁₆≤δ_(HV)<2π

where:

A₁₃=β₁−π−φ₂

A₁₆=β₁+φ₂

This region is identified by the horizontal cross-hatching

FIG. 15. These conditions can be defined in one mathematical expressionas:

$L_{Q_{2}} = {{{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - 0 + \sigma}{{{\delta_{HV} - 0}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{13} - \delta_{HV} - \sigma}{{{A_{13} - \delta_{HV}}} + \sigma} + 1} \right\rbrack} + {{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{16} + \sigma}{{{\delta_{HV} - A_{16}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{{2\pi} - \delta_{HV} - \sigma}{{{{2\pi} - \delta_{HV}}} + \sigma} + 1} \right\rbrack}}$

and the remote vehicle 14 is to the right (VU=10) of the host vehicle 10if:A ₁₄≤δ_(HV) <A ₁₅

where:

A₁₄=β₁−π+φ₂

A₁₅=β₁−φ₂

This region is identified by the checkered cross-hatching

in FIG. 15. These conditions can be defined in one mathematicalexpression as:

$R_{Q_{2}} = {{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{14} + \sigma}{{{\delta_{HV} - A_{14}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{15} - \delta_{HV} - \sigma}{{{A_{15} - \delta_{HV}}} + \sigma} + 1} \right\rbrack}$

The expressions are then consolidated in the following matrix for thecase when the remote vehicle 14 is to the northwest of the host vehicle10.

Q₂ Lateral Position RV in lane (I_(Q) ₂ ) RV Left (L_(Q) ₂ ) RV Right(R_(Q) ₂ ) Unused Longitudinal RV Ahead (P_(Q2)) Q₂ × P_(Q) ₂ × I_(Q) ₂Q₂ × P_(Q) ₂ × L_(Q) ₂ Q₂ × P_(Q) ₂ × R_(Q) ₂ 0 Position RV Adjacent(A_(Q2)) Q₂ × A_(Q) ₂ × I_(Q) ₂ Q₂ × A_(Q) ₂ × L_(Q) ₂ Q₂ × A_(Q) ₂ ×R_(Q) ₂ 0 RV Behind (B_(Q2)) Q₂ × B_(Q) ₂ × I_(Q) ₂ Q₂ × B_(Q) ₂ × L_(Q)₂ Q₂ × B_(Q) ₂ × R_(Q) ₂ 0 Unused 0 0 0 0

FIGS. 16 and 17 illustrate a condition in which the remote vehicle 14 isto the Northeast of the host vehicle 10, and is in quadrant Q3.

Q3: Remote Vehicle 14 is to the Southwest of the Host Vehicle 10

$Q_{3} = {{\frac{1}{4}\left\lbrack {\frac{\phi_{HV} - \phi_{RV} - \sigma}{{{\phi_{HV} - \phi_{RV}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{\theta_{HV} - \theta_{RV} + \sigma}{{{\theta_{HV} - \theta_{RV}}} + \sigma} + 1} \right\rbrack}$

If the remote vehicle 14 is southwest of the host vehicle 10 as shown inFIGS. 16 and 17, both latitude and longitude for the remote vehicle 14is less than the latitude and longitude for the host vehicle 10. Underthese conditions, the expression for Q3 above will equal 1 otherwise itwill equal 0.

Longitudinal Position (XW)

The remote vehicle 14 is ahead (XW=00) of the host vehicle 10 if:A ₁₂≤δ_(HV) <A ₁

where:

A₁₂=β₁π/2+φ₁

A₁=β₁+π2−φ₁

φ₁ is a threshold value that defines the angular range in which theremote vehicle 14 is defined to be adjacent to the host vehicle 10.

This region β₁ calculated by the following equation is identified by thevertical cross-hatching

in FIG. 16.

$\beta_{1} = {{\pi\left\lbrack {\frac{\theta_{HV} - \theta_{RV} - \sigma}{{{\theta_{HV} - \theta_{RV}}} + \sigma} + 1} \right\rbrack} - {{\cos^{- 1}\left( \frac{\left( {\phi_{RV} - \phi_{HV}} \right)}{\sqrt{{\left( {\theta_{RV} - \theta_{HV}} \right)^{2}\cos^{2}\phi_{HV}} + \left( {\phi_{RV} - \phi_{HV}} \right)^{2}}} \right)}\left\lbrack \frac{\theta_{HV} - \theta_{RV} - \sigma}{{{\theta_{HV} - \theta_{RV}}} + \sigma} \right\rbrack}}$

These conditions can be defined in one mathematical expression as:

$P_{Q_{3}} = {{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{12} + \sigma}{{{\delta_{HV} - A_{12}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{1} - \delta_{HV} - \sigma}{{{A_{1} - \delta_{HV}}} + \sigma} + 1} \right\rbrack}$

and the remote vehicle 14 is adjacent (XW=01) to the host vehicle 10 if:A ₁≤δ_(HV) <A ₂ or A ₁₁≤δ_(HV) <A ₁₂

where:

A₁=β₁+π/2+_(φ) 1

A₂=β₁+π/2+φ₁

A₁₁=β₁−π/2−φ₁

A₁₂=β₁−π/2+φ₁

These two specific angular ranges are identified by the checkeredcross-hatching

as the interfaces between the area identified by the verticalcross-hatching

and the area identified by the slanted cross-hatching

in FIG. 16.

These conditions can be defined in one mathematical expression as:

$A_{Q_{3}} = {{{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{1} + \sigma}{{{\delta_{HV} - A_{1}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{2} - \delta_{HV} - \sigma}{{{A_{2} - \delta_{HV}}} + \sigma} + 1} \right\rbrack} + {{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{11} + \sigma}{{{\delta_{HV} - A_{11}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{12} - \delta_{HV} - \sigma}{{{A_{12} - \delta_{HV}}} + \sigma} + 1} \right\rbrack}}$

and the remote vehicle 14 is behind (XW=10) the host vehicle 10 if:0≤δ_(HV) <A ₁₁ or A ₂≤δ_(HV)<2π

where:

A₂=β₁+π/2+φ₁

A₁₁=β₂−π/2−φ₁

This region is identified by the slanted cross-hatching

in FIG. 16. These conditions can be defined in one mathematicalexpression as:

$B_{Q_{3}} = {{{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - 0 + \sigma}{{{\delta_{HV} - 0}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{11} - \delta_{HV} - \sigma}{{{A_{11} - \delta_{HV}}} + \sigma} + 1} \right\rbrack} + {{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{2} + \sigma}{{{\delta_{HV} - A_{2}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{{2\pi} - \delta_{HV} - \sigma}{{{{2\pi} - \delta_{HV}}} + \sigma} + 1} \right\rbrack}}$

Lateral Position (VU)

The remote vehicle 14 is in lane (VU=00) with the host vehicle 10 if:A ₁₃≤δ_(HV) ≤A ₁₄ or A ₁₅≤δ_(HV) <A ₁₆

where:

A₁₃=β₁−π−φ₂

A₁₄=β₁−π+φ₂

A₁₅=β₁−φ₂

A₁₆=β₁+φ₂

φ₂ on is a threshold value that defines the angular range in which theRV is defined to be in the same lane with the HV.

These two specific angular ranges are identified by the verticalcross-hatching

as the interfaces between area identified by the checkeredcross-hatching

and the area identified by the horizontal cross-hatching

FIG. 17. These conditions can be defined in one mathematical expressionas:

$I_{Q_{3}} = {{{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{13} + \sigma}{{{\delta_{HV} - A_{13}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{14} - \delta_{HV} - \sigma}{{{A_{14} - \delta_{HV}}} + \sigma} + 1} \right\rbrack} + {{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{15} + \sigma}{{{\delta_{HV} - A_{15}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{16} - \delta_{HV} - \sigma}{{{A_{16} - \delta_{HV}}} + \sigma} + 1} \right\rbrack}}$

and the remote vehicle 14 is to the left (VU=01) of the host vehicle 10if:0≤δ_(HV) <A ₁₃ or A ₁₆≤δ_(HV)<2π

A₁₃=β₁−π−φ₂

A₁₆=β₁+φ₂

This region is identified by the horizontal cross-hatching

FIG. 17. These conditions can be defined in one mathematical expressionas:

$L_{Q_{3}} = {{{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - 0 + \sigma}{{{\delta_{HV} - 0}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{13} - \delta_{HV} - \sigma}{{{A_{13} - \delta_{HV}}} + \sigma} + 1} \right\rbrack} + {{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{16} + \sigma}{{{\delta_{HV} - A_{16}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{{2\pi} - \delta_{HV} - \sigma}{{{{2\pi} - \delta_{HV}}} + \sigma} + 1} \right\rbrack}}$

and the remote vehicle 14 is to the right (VU=10) of the host vehicle 10if:A ₁₄≤δ_(HV) <A ₁₅

where:

A₁₄=β₁−π+φ₂

A₁₅=β₁−φ₂

This region is identified by the checkered cross-hatching

in FIG. 17. These conditions can be defined in one mathematicalexpression as:

$R_{Q_{3}} = {{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{14} + \sigma}{{{\delta_{HV} - A_{14}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{15} - \delta_{HV} - \sigma}{{{A_{15} - \delta_{HV}}} + \sigma} + 1} \right\rbrack}$

The expressions are then consolidated in the following matrix for thecase when the remote vehicle 14 is to the southwest of the host vehicle10.

Q₃ Lateral Position RV in lane (I_(Q) ₃ ) RV Left (L_(Q) ₃ ) RV Right(R_(Q) ₃ ) Unused Longitudinal RV Ahead (P_(Q) ₃ ) Q₃ × P_(Q) ₃ × I_(Q)₃ Q₃ × P_(Q) ₃ × L_(Q) ₃ Q₃ × P_(Q) ₃ × R_(Q) ₃ 0 Position RV Adjacent(A_(Q) ₃ ) Q₃ × A_(Q) ₃ × I_(Q) ₃ Q₃ × A_(Q) ₃ × L_(Q) ₃ Q₃ × A_(Q) ₃ ×R_(Q) ₃ 0 RV Behind (B_(Q) ₃ ) Q₃ × B_(Q) ₃ × I_(Q) ₃ Q₃ × B_(Q) ₃ ×L_(Q) ₃ Q₃ × B_(Q) ₃ × R_(Q) ₃ 0 Unused 0 0 0 0

FIGS. 18 and 19 illustrate a condition in which the remote vehicle 14 isto the Southeast of the host vehicle 10, and is in quadrant Q4.

Q4: Remote Vehicle 14 is to the Southeast of the Host Vehicle 10

$Q_{4} = {{\frac{1}{4}\left\lbrack {\frac{\phi_{HV} - \phi_{RV} + \sigma}{{{\phi_{HV} - \phi_{RV}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{\theta_{RV} - \phi_{HV} - \sigma}{{{\theta_{RV} - \theta_{HV}}} + \sigma} + 1} \right\rbrack}$

If the remote vehicle 14 is southeast of the host vehicle 10 as shown inthe FIGS. 18 and 19, the latitude for the remote vehicle 14 is less thanthe latitude of the host vehicle 10 but the longitude for the remotevehicle 14 is greater than the longitude for the host vehicle 10. Underthese conditions, the expression for Q4 above will equal 1 otherwise itwill equal 0.

Longitudinal Position (XW)

The remote vehicle 14 is ahead (XW=00) of the host vehicle 10 if:A ₁₂≤δ_(HV) <A ₁

where:

A₁=β₁+π/2−φ₁

A₁₂=β₁−π/2+φ₁

φ₁ is a threshold value that defines the angular range in which theremote vehicle 14 is defined to be adjacent to the host vehicle 10.

This region β₁ calculated by the following equation is identified by thevertical cross-hatching

in FIG. 18.

$\beta_{1} = {{\pi\left\lbrack {\frac{\theta_{HV} - \theta_{RV} - \sigma}{{{\theta_{HV} - \theta_{RV}}} + \sigma} + 1} \right\rbrack} - {{\cos^{- 1}\left( \frac{\left( {\phi_{RV} - \phi_{HV}} \right)}{\sqrt{{\left( {\theta_{RV} - \theta_{HV}} \right)^{2}\cos^{2}\phi_{HV}} + \left( {\phi_{RV} - \phi_{HV}} \right)^{2}}} \right)}\left\lbrack \frac{\theta_{HV} - \theta_{RV} - \sigma}{{{\theta_{HV} - \theta_{RV}}} + \sigma} \right\rbrack}}$

These conditions can be defined in one mathematical expression as:

$P_{Q_{4}} = {{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{12} + \sigma}{{{\delta_{HV} - A_{12}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{1} - \delta_{HV} - \sigma}{{{A_{1} - \delta_{HV}}} + \sigma} + 1} \right\rbrack}$

and the remote vehicle 14 is adjacent (XW=01) to the host vehicle 10 if:A ₁≤δ_(HV) <A ₂ or A ₁₁≤δ_(HV) <A ₁₂

where:

A₁=β₁+π/2−φ₁

A₂=β₁+π/2+φ₁

A₁₁=β₁−π/2−φ₁

A₁₂=β₁−π/2+φ₁

These two specific angular ranges are identified by the checkeredcross-hatching

as the interfaces between area identified by the vertical cross-hatching

and the slanted cross-hatching

in FIG. 18. These conditions can be defined in one mathematicalexpression as:

$A_{Q_{4}} = {{{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{1} + \sigma}{{{\delta_{HV} - A_{1}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{2} - \delta_{HV} - \sigma}{{{A_{2} - \delta_{HV}}} + \sigma} + 1} \right\rbrack} + {{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{11} + \sigma}{{{\delta_{HV} - A_{11}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{12} - \delta_{HV} - \sigma}{{{A_{12} - \delta_{HV}}} + \sigma} + 1} \right\rbrack}}$

and the remote vehicle 14 is behind (XW=10) the host vehicle 10 if:A ₂≤δ_(HV)<2π or 0≤δ_(HV) <A ₁₁

where:

A₂=β₁+π/2+φ₁

A₁₁=β₁−π/2−φ₁

This region is identified by the slanted cross-hatching

in FIG. 18. These conditions can be defined in one mathematicalexpression as:

$B_{Q_{4}} = {{{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - 0 + \sigma}{{{\delta_{HV} - 0}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{11} - \delta_{HV} - \sigma}{{{A_{11} - \delta_{HV}}} + \sigma} + 1} \right\rbrack} + {{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{2} + \sigma}{{{\delta_{HV} - A_{2}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{{2\pi} - \delta_{HV} - \sigma}{{{{2\pi} - \delta_{HV}}} + \sigma} + 1} \right\rbrack}}$

Lateral Position (VU)

The remote vehicle 14 is in lane (VU=00) with the host vehicle 10 if:A ₅≤δ_(HV) <A ₆ or A ₇≤δ_(HV) <A ₈

where:

A₅=β₁−φ₂

A₆=β₁+φ₂

A₇=β₁+π+φ₂

A₈=β₁+π+φ₂

φ₂ is a threshold value that defines the angular range in which theremote vehicle 14 is defined to be in the same lane with the hostvehicle 10.

These two specific angular ranges are identified by the verticalcross-hatching 11111 as the interfaces between area identified by thecheckered cross-hatching

and the horizontal cross-hatching

in FIG. 19. These conditions can be defined in one mathematicalexpression as:

$I_{Q_{4}} = {{{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{5} + \sigma}{{{\delta_{HV} - A_{5}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{6} - \delta_{HV} - \sigma}{{{A_{6} - \delta_{HV}}} + \sigma} + 1} \right\rbrack} + {{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{7} + \sigma}{{{\delta_{HV} - A_{7}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{8} - \delta_{HV} - \sigma}{{{A_{8} - \delta_{HV}}} + \sigma} + 1} \right\rbrack}}$

and the remote vehicle 14 is to the left (VU=01) of the host vehicle 10if:A ₆≤δ_(HV) <A ₇

where:

A₆=β₁+φ₂

A₇=β₁+π−₂

This region is identified as the horizontal cross-hatching

in FIG. 19. These conditions can be defined in one mathematicalexpression as:

$L_{Q_{4}} = {{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{6} + \sigma}{{{\delta_{HV} - A_{6}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{7} - \delta_{HV} - \sigma}{{{A_{7} - \delta_{HV}}} + \sigma} + 1} \right\rbrack}$

and the remote vehicle 14 is to the right (VU=10) of the host vehicle 10if:0≤δ_(HV) <A ₅ or A ₈≤δ_(HV)<2π

where:

A₅=β₁−φ₂

A₈=β₁+π+φ₂

This region is identified by the checkered cross-hatching

in FIG. 19. These conditions can be defined in one mathematicalexpression as:

$R_{Q_{4}} = {{{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - 0 + \sigma}{{{\delta_{HV} - 0}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{5} - \delta_{HV} - \sigma}{{{A_{5} - \delta_{HV}}} + \sigma} + 1} \right\rbrack} + {{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{8} + \sigma}{{{\delta_{HV} - A_{8}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{{2\pi} - \delta_{HV} - \sigma}{{{{2\pi} - \delta_{HV}}} + \sigma} + 1} \right\rbrack}}$

The expressions are then consolidated in the following matrix for thecase when the remote vehicle 14 is to the southwest of the host vehicle10.

Q₄ Lateral Position RV in lane (I_(Q) ₄ ) RV Left (L_(Q) ₄ ) RV Right(R_(Q) ₄ ) Unused Longitudinal RV Ahead (P_(Q) ₄ ) Q₄ × P_(Q) ₄ × I_(Q)₄ Q₄ × P_(Q) ₄ × L_(Q) ₄ Q₄ × P_(Q) ₄ × R_(Q) ₄ 0 Position RV Adjacent(A_(Q) ₄ ) Q₄ × A_(Q) ₄ × I_(Q) ₄ Q₄ × A_(Q) ₄ × L_(Q) ₄ Q₄ × A_(Q) ₄ ×R_(Q) ₄ 0 RV Behind (B_(Q) ₄ ) Q₄ × B_(Q) ₄ × I_(Q) ₄ Q₄ × B_(Q) ₄ ×L_(Q) ₄ Q₄ × B_(Q) ₄ × R_(Q) ₄ 0 Unused 0 0 0 0

The following is a Summary for the four Quadrants Q1 through Q4:

Q₁ Lateral Position RV in lane (I_(Q) ₁ ) RV Left (L_(Q) ₁ ) RV Right(R_(Q) ₁ ) Unused Longitudinal RV Ahead (P_(Q) ₁ ) Q₁ × P_(Q) ₁ × I_(Q)₁ Q₁ × P_(Q) ₁ × L_(Q) ₁ Q₁ × P_(Q) ₁ × R_(Q) ₁ 0 Position RV Adjacent(A_(Q) ₁ ) Q₁ × A_(Q) ₁ × I_(Q) ₁ Q₁ × A_(Q) ₁ × L_(Q) ₁ Q₁ × A_(Q) ₁ ×R_(Q) ₁ 0 RV Behind (B_(Q) ₁ ) Q₁ × B_(Q) ₁ × I_(Q) ₁ Q₁ × B_(Q) ₁ ×L_(Q) ₁ Q₁ × B_(Q) ₁ × R_(Q) ₁ 0 Unused 0 0 0 0 Q₂ Lateral Position RVin lane (I_(Q) ₂ ) RV Left (L_(Q) ₂ ) RV Right (R_(Q) ₂ ) UnusedLongitudinal RV Ahead (P_(Q2)) Q₂ × P_(Q) ₂ × I_(Q) ₂ Q₂ × P_(Q) ₂ ×L_(Q) ₂ Q₂ × P_(Q) ₂ × R_(Q) ₂ 0 Position RV Adjacent (A_(Q2)) Q₂ ×A_(Q) ₂ × I_(Q) ₂ Q₂ × A_(Q) ₂ × L_(Q) ₂ Q₂ × A_(Q) ₂ × R_(Q) ₂ 0 RVBehind (B_(Q2)) Q₂ × B_(Q) ₂ × I_(Q) ₂ Q₂ × B_(Q) ₂ × L_(Q) ₂ Q₂ × B_(Q)₂ × R_(Q) ₂ 0 Unused 0 0 0 0 Q₃ Lateral Position RV in lane (I_(Q) ₃ )RV Left (L_(Q) ₃ ) RV Right (R_(Q) ₃ ) Unused Longitudinal RV Ahead(P_(Q) ₃ ) Q₃ × P_(Q) ₃ × I_(Q) ₃ Q₃ × P_(Q) ₃ × L_(Q) ₃ Q₃ × P_(Q) ₃ ×R_(Q) ₃ 0 Position RV Adjacent (A_(Q) ₃ ) Q₃ × A_(Q) ₃ × I_(Q) ₃ Q₃ ×A_(Q) ₃ × L_(Q) ₃ Q₃ × A_(Q) ₃ × R_(Q) ₃ 0 RV Behind (B_(Q) ₃ ) Q₃ ×B_(Q) ₃ × I_(Q) ₃ Q₃ × B_(Q) ₃ × L_(Q) ₃ Q₃ × B_(Q) ₃ × R_(Q) ₃ 0 Unused0 0 0 0 Q₄ Lateral Position RV in lane (I_(Q) ₄ ) RV Left (L_(Q) ₄ ) RVRight (R_(Q) ₄ ) Unused Longitudinal RV Ahead (P_(Q) ₄ ) Q₄ × P_(Q) ₄ ×I_(Q) ₄ Q₄ × P_(Q) ₄ × L_(Q) ₄ Q₄ × P_(Q) ₄ × R_(Q) ₄ 0 Position RVAdjacent (A_(Q) ₄ ) Q₄ × A_(Q) ₄ × I_(Q) ₄ Q₄ × A_(Q) ₄ × L_(Q) ₄ Q₄ ×A_(Q) ₄ × R_(Q) ₄ 0 RV Behind (B_(Q) ₄ ) Q₄ × B_(Q) ₄ × I_(Q) ₄ Q₄ ×B_(Q) ₄ × L_(Q) ₄ Q₄ × B_(Q) ₄ × R_(Q) ₄ 0 Unused 0 0 0 0

The longitudinal relative position bits XW and the lateral relativeposition bits VU for the relative position code are defined as follows:

VU 00 01 10 11 XW 00 0000 0001 0010 0011 01 0100 0101 0110 0111 10 10001001 1010 1011 11 1100 1101 1110 1111

Bits X through U are generated using the following array of expressions.

X w v u x₁ = 0 w₁ = 0 v₁ = 0 u₁ = 0 x₂ = 0 w₂ = 0 v₂ = 0$u_{2} = {\sum\limits_{i = 1}^{4}{Q_{i} \times P_{Q_{i}} \times L_{Q_{i}} \times 1}}$x₃ = 0 w₃ = 0$v_{3} = {\sum\limits_{i = 1}^{4}{Q_{i} \times P_{Q_{i}} \times R_{Q_{i}} \times 1}}$u₃ = 0 x₄ = 0$w_{4} = {\sum\limits_{i = 1}^{4}{Q_{i} \times A_{Q_{i}} \times I_{Q_{i}} \times 1}}$v₄ = 0 u₄ = 0 x₅ = 0$w_{5} = {\sum\limits_{i = 1}^{4}{Q_{i} \times A_{Q_{i}} \times L_{Q_{i}} \times 1}}$v₅ = 0$u_{5} = {\sum\limits_{i = 1}^{4}{Q_{i} \times A_{Q_{i}} \times L_{Q_{i}} \times 1}}$x₆ = 0$w_{6} = {\sum\limits_{i = 1}^{4}{Q_{i} \times A_{Q_{i}} \times R_{Q_{i}} \times 1}}$$v_{6} = {\sum\limits_{i = 1}^{4}{Q_{i} \times A_{Q_{i}} \times R_{Q_{i}} \times 1}}$u₆ = 0$x_{7} = {\sum\limits_{i = 1}^{4}{Q_{i} \times B_{Q_{i}} \times I_{Q_{i}} \times 1}}$w₇ = 0 v₇ = 0 u₇ = 0$x_{8} = {\sum\limits_{i = 1}^{4}{Q_{i} \times B_{Q_{i}} \times I_{Q_{i}} \times 1}}$w₈ = 0 v₈ =$u_{8} = {\sum\limits_{i = 1}^{4}{Q_{i} \times B_{Q_{i}} \times L_{Q_{i}} \times 1}}$$x_{9} = {\sum\limits_{i = 1}^{4}{Q_{i} \times B_{Q_{i}} \times R_{Q_{i}} \times 1}}$w₉ = 0$v_{9} = {\sum\limits_{i = 1}^{4}{Q_{i} \times B_{Q_{i}} \times R_{Q_{i}} \times 1}}$u₉ = 0 $X = {\sum\limits_{i = 1}^{9}x_{i}}$$W = {\sum\limits_{i = 1}^{9}w_{i}}$$V = {\sum\limits_{i = 1}^{9}v_{i}}$$U = {\sum\limits_{i = 1}^{9}u_{i}}$

Elevation

The elevation component of relative position is provided by thefollowing three expressions.

If the host vehicle 10 and the remote vehicle 14 are at the sameelevation,

$Z_{1} = {{{\frac{1}{4}\left\lbrack {\frac{ɛ - \left( {z_{HV} - z_{RV}} \right) + \sigma}{{{ɛ - \left( {z_{HV} - z_{RV}} \right)}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{ɛ - \left( {z_{RV} - z_{HV}} \right) - \sigma}{{{ɛ - \left( {z_{RV} - z_{HV}} \right)}} + \sigma} + 1} \right\rbrack} = {1\left( {{TS} = 00} \right)}}$${{If}\mspace{14mu}{the}\mspace{14mu}{HV}\mspace{14mu}{is}\mspace{14mu}{lower}},{Z_{2} = {{\frac{1}{2}\left\lbrack {\frac{\left( {z_{RV} - z_{HV}} \right) - ɛ - \sigma}{{{\left( {z_{RV} - z_{HV}} \right) - ɛ}} + \sigma} + 1} \right\rbrack} = {1\left( {{TS} = 01} \right)}}}$${{If}\mspace{14mu}{the}\mspace{14mu}{host}\mspace{14mu}{vehicle}\mspace{14mu} 10\mspace{14mu}{is}\mspace{14mu}{higher}},{Z_{3} = {{\frac{1}{2}\left\lbrack {\frac{\left( {z_{HV} - z_{RV}} \right) - ɛ - \sigma}{{{\left( {z_{HV} - z_{RV}} \right) - ɛ}} + \sigma} + 1} \right\rbrack} = {1\left( {{TS} = 10} \right)}}}$where:z_(HV)=HV elevationz_(RV)=RV elevationε=a defined threshold value of distance such as 4 m.

Bits T and S are generated using the following array of expressions.

t s t₁ = Z₁ × 0 s₁ = Z₁ × 0 t₂ = Z₂ × 0 s₂ = Z₂ × 1 t₃ = z₃ × 1 s₃ = Z₃× 0 $T = {\sum\limits_{i = 1}^{3}t_{i}}$$S = {\sum\limits_{i = 1}^{3}s_{i}}$

Remote Vehicle Position Relative to Host Vehicle (Heading)

HV and RV traveling in same direction (RQ=01)

Remote Vehicle Heading angle as a function of Host Vehicle heading anglefor the case of following vehicles can be defined as follows:δ_(RV)=δ_(HV)

However, narrowly defining δ_(RV) to be exactly the same as δ_(HV) wouldresult in a condition where the two vehicles would almost never beclassified as heading in the same direction when in reality thiscondition is a very common occurrence. In order to account for smalldifferences in heading angles, a variable φ₂ is used to define a rangeof heading angles for the RV in which the RV would be considered to beheading in the same direction as the HV. To define this range, thefollowing expressions are defined:

Minimum RV heading angle

If σ_(RV)−φ₂<0 then δ_(RV) _(min) ⁰¹=2π+δ_(RV)−φ₂

If δ_(RV)−φ₂≥0 then δ_(RV) _(min) ⁰¹=δ_(RV)−φ₂

These conditions can be combined into one mathematical expression as:δ_(RV) _(min) ⁰¹=ζ_(min) ₁ ×(2π+δ_(RV)−φ₂)+ζ_(min) ₁ ×(δ_(RV)−φ₂)where:

$\zeta_{\min_{1}} = {\frac{1}{2}\left\lbrack {\frac{0 - \left( {\delta_{RV} - \varphi_{2}} \right) - \sigma}{{{0 - \left( {\delta_{RV} - \varphi_{2}} \right)}} + \sigma} + 1} \right\rbrack}$$\zeta_{\min_{2}} = {\frac{1}{2}\left\lbrack {\frac{\left( {\delta_{RV} - \varphi} \right) - 0 + \sigma}{{{\left( {\delta_{RV} - \varphi} \right) - 0}} + \sigma} + 1} \right\rbrack}$

These expressions have two values, 0 or 1 depending on the value ofδ_(RV) and can be thought of as filtering functions that ensure theappropriate expression is used to calculate the value of δ_(RV) _(min)⁰¹.

Maximum RV heading angle

If δ_(RV)+φ<2π then δ_(RV) _(max) ⁰¹=δ_(RV)+φ₂

If δ_(RV)+φ≥2π then δ_(RV) _(max) ⁰¹=δ_(RV)+φ₂−2π

These conditions can be combined into one mathematical expression as:δ_(RV) _(max) ⁰¹=ζ_(max) ₁ ×(δ_(RV)+φ₂)+ζ_(max) ₂ ×(δ_(RV)+φ₂−2π)

where:

$\zeta_{\max_{1}} = {\frac{1}{2}\left\lbrack {\frac{{2\pi} - \left( {\delta_{RV} + \varphi_{2}} \right) - \sigma}{{{{2\pi} - \left( {\delta_{RV} + \varphi_{2}} \right)}} + \sigma} + 1} \right\rbrack}$$\zeta_{\max_{2}} = {\frac{1}{2}\left\lbrack {\frac{\left( {\delta_{RV} + \varphi_{2}} \right) - {2\pi} + \sigma}{{{\left( {\delta_{RV} + \varphi_{2}} \right) - {2\pi}}} + \sigma} + 1} \right\rbrack}$

These expressions have two values, 0 or 1 depending on the value ofδ_(RV) and can be thought of as filtering functions that ensure theappropriate expression is used to calculate the value of δ_(RV) _(max)⁰¹.

The remote vehicle 14 is considered to be traveling in the samedirection as the host vehicle 10 when the heading angle of the remotevehicle 14, δ_(RV) falls within the range δ_(RV) _(min) ⁰¹ and δ_(RV)_(max) ⁰¹ therefore in most cases, the heading angle of the host vehicle10, δ_(HV) will be greater than or equal to δ_(RV) _(min) ⁰¹ and lessthan or equal to δ_(RV) _(max) ⁰¹ otherwise the remote vehicle 14 willbe considered to be traveling in a direction other than the samedirection of the HV as shown in FIGS. 20-23 which represented δ_(RV)_(min) ⁰¹≤δ_(HV)<δ_(RV) _(max) ⁰¹.

However, because of the fixed reference used where North=0°, there arecases where δ_(HV) will be less than or equal to δ_(RV) _(min) ⁰¹ andless than or equal to δ_(RV) _(max) ⁰¹ or cases where δ_(HV) will begreater than or equal to δ_(RV) _(min) ⁰¹ and greater than or equal toδ_(RV) _(max) ⁰¹ such as shown in FIGS. 24 and 25. In FIG. 24, δ_(HV)less than δ_(RV) _(min) ⁰¹ and less than δ_(RV) _(max) ⁰¹. In FIG. 25,δ_(HV) greater than δ_(RV) _(min) ⁰¹ and greater than δ_(RV) _(max) ⁰¹.

Consider the following expressions for H₁ and H₂:

H₁=δ_(HV)−δ_(RV) _(min) ⁰¹

H₂=δ_(HV)−δ_(RV) _(max) ⁰¹

For any value of δ_(HV), the values for H₁ and H₂ fall within threedistinct categories:

-   -   1: H₁ is negative, H₂ is negative and H₁<H₂ (δ_(HV)<δ_(RV)        _(min) ⁰¹ and δ_(HV)<δ_(RV) _(max) ⁰¹)    -   2: H₁ is positive, H₂ is negative and H₁>H₂ (δ_(HV)>δ_(RV)        _(min) ⁰¹ and δ_(HV)<δ_(RV) _(max) ⁰¹)    -   3: H₁ is positive, H₂ is positive and H₁<H₂ (δ_(HV)>δ_(RV)        _(min) ⁰¹ and δ_(RV) _(max) ⁰¹)

From these three conditions, it can be shown that for any combination ofδ_(HV) and δ_(RV), where 0≤δ_(HV)<2π and 0≤δ_(RV)<2π the followingexpressions can be used to identify if the HV and RV are traveling inthe same direction.

$\Delta_{1}^{01} = {{\frac{1}{8}\left\lbrack {\frac{\delta_{{RV}_{\min}}^{01} - \delta_{RV} + \sigma}{{{\delta_{{RV}_{\min}}^{01} - \delta_{RV}}} + \sigma} + 1} \right\rbrack} \times {\quad\left\lbrack {{{\left. \quad{\frac{\delta_{{RV}_{\max}}^{01} - \delta_{RV} + \sigma}{{{\delta_{{RV}_{\max}}^{01} - \delta_{RV}}} + \sigma} + 1} \right\rbrack \times \left\lbrack {1 - \frac{H_{1} - H_{2} - \sigma}{{{H_{1} - H_{2}}} + \sigma}} \right\rbrack{If}\mspace{14mu} H_{1}} < H_{2}},{{\delta_{RV} \leq {\Delta_{{RV}_{\min}}^{01}{and}\mspace{14mu}\delta_{RV}} \leq {\delta_{{RV}_{\max}}^{01}\Delta_{1}^{01}}} = {{1\mspace{14mu}{otherwise}\mspace{14mu}\Delta_{1}^{01}} = {{0\Delta_{2}^{01}} = {{\frac{1}{8}\left\lbrack {\frac{\delta_{RV} - \delta_{{RV}_{\min}}^{01} + \sigma}{{{\delta_{RV} - \delta_{{RV}_{\min}}^{01}}} + \sigma} + 1} \right\rbrack} \times {\quad{{\left\lbrack {\frac{\delta_{{RV}_{\max}}^{01} - \delta_{RV} + \sigma}{{{\delta_{{RV}_{\max}}^{01} - \delta_{RV}}} + \sigma} + 1} \right\rbrack \times \left\lbrack {\frac{H_{1} - H_{2} - \sigma}{{{H_{1} - H_{2}}} + \sigma} + 1} \right\rbrack{{{If}\mspace{14mu} H_{1}} > {H_{2}\mspace{14mu}{and}\mspace{14mu}\delta_{{RV}_{\min}}^{01}} \leq \delta_{RV} \leq_{{RV}_{\max}}^{01}}},{\Delta_{2}^{01} = {{1\mspace{14mu}{otherwise}\mspace{14mu}\Delta_{2}^{01}} = {{0\Delta_{3}^{01}} = {{\frac{1}{8}\left\lbrack {\frac{\delta_{RV} - \delta_{{RV}_{\min}}^{01} + \sigma}{{{\delta_{RV} - \delta_{{RV}_{\min}}^{01}}} + \sigma} + 1} \right\rbrack} \times {\quad{\left\lbrack {\frac{\delta_{RV} - \delta_{{RV}_{\max}}^{01} + \sigma}{{{\delta_{RV} - \delta_{{RV}_{\max}}^{01}}} + \sigma} + 1} \right\rbrack \times \left\lbrack {1 - \frac{H_{1} - H_{2} - \sigma}{{{H_{1} - H_{2}}} + \sigma}} \right\rbrack}}}}}}}}}}}}} \right.}}$

If H₁<H₂ and δ_(RV) _(min) ⁰¹≤δ_(RV) and δ_(RV) _(max) ⁰¹≤δ_(RV) Δ₁ ⁰¹=1otherwise Δ₁ ⁰¹=0

Also, it is advantageous to define the difference of H₁ and H₂ asfollows:H ₁ −H ₂=δ_(RV)−δ_(RV) _(min) ⁰¹−(δ_(HV)−δ_(RV) _(max) ⁰¹)H ₁ −H ₂=δ_(HV)−δ_(RV) _(min) ⁰¹−δ_(HV)+δ_(RV) _(max) ⁰¹H ₁ −H ₂=δ_(HV)−δ_(RV) _(min) ⁰¹−δ_(HV)+δ_(RV) _(max) ⁰¹H ₁ −H ₂=δ_(RV) _(max) ⁰¹−δ_(RV) _(min) ⁰¹

Then the previous expressions can be expressed as:

$\Delta_{1}^{01} = {{\frac{1}{8}\left\lbrack {\frac{\delta_{{RV}_{\min}}^{01} - \delta_{RV} + \sigma}{{{\delta_{{RV}_{\min}}^{01} - \delta_{RV}}} + \sigma} + 1} \right\rbrack} \times {\quad{{\left\lbrack {\frac{\delta_{{RV}_{\max}}^{01} - \delta_{RV} + \sigma}{{{\delta_{{RV}_{\max}}^{01} - \delta_{RV}}} + \sigma} + 1} \right\rbrack \times \left\lbrack {1 - \frac{\delta_{{RV}_{\max}}^{01} - \delta_{{RV}_{\min}}^{01} - \sigma}{{{\delta_{{RV}_{\max}}^{01} - \delta_{{RV}_{\min}}^{01}}} + \sigma}} \right\rbrack\Delta_{2}^{01}} = {{\frac{1}{8}\left\lbrack {\frac{\delta_{RV} - \delta_{{RV}_{\min}}^{01} + \sigma}{{{\delta_{RV} - \delta_{{RV}_{\min}}^{01}}} + \sigma} + 1} \right\rbrack} \times {\quad{{\left\lbrack {\frac{\delta_{{RV}_{\max}}^{01} - \delta_{RV} + \sigma}{{{\delta_{{RV}_{\max}}^{01} - \delta_{RV}}} + \sigma} + 1} \right\rbrack \times \left\lbrack {\frac{\delta_{{RV}_{\max}}^{01} - \delta_{{RV}_{\min}}^{01} - \sigma}{{{\delta_{{RV}_{\max}}^{01} - \delta_{{RV}_{\min}}^{01}}} + \sigma} + 1} \right\rbrack\Delta_{3}^{01}} = {{\frac{1}{8}\left\lbrack {\frac{\delta_{RV} - \delta_{{RV}_{\min}}^{01} + \sigma}{{{\delta_{RV} - \delta_{{RV}_{\min}}^{01}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{\delta_{RV} - \delta_{{RV}_{\max}}^{01} + \sigma}{{{\delta_{RV} - \delta_{{RV}_{\max}}^{01}}} + \sigma} + 1} \right\rbrack \times {\quad\left\lbrack {{- 1}\frac{\delta_{{RV}_{\max}}^{01} - \delta_{{RV}_{\min}}^{01} - \sigma}{{{\delta_{{RV}_{\max}}^{01} - \delta_{{RV}_{\min}}^{01}}} + \sigma}} \right\rbrack}}}}}}}}$

If the sum of these three expressions is equal to 1, the host vehicle 10and the remote vehicle 14 are traveling in the same direction. Thiscondition is expressed mathematically as:

${\sum\limits_{i = 1}^{3}\Delta_{i}^{01}} = {1\left( {{RQ} = 01} \right)}$thus:

$r_{1} = {\sum\limits_{i = 1}^{3}{\Delta_{i}^{01} \times 0}}$$q_{1} = {\sum\limits_{i = 1}^{3}{\Delta_{i}^{01} \times 1}}$

Host Vehicle and Remote Vehicle approaching either other from oppositedirections (RQ=10):

Remote Vehicle Heading angle as a function of Host Vehicle heading anglefor the case of on-coming vehicles can be defined as follows:

$\delta_{RV} = {{{\frac{1}{2}\left\lbrack {\frac{\delta_{HV} - \pi - \sigma}{{{\delta_{HV} - \pi}} + \sigma} + 1} \right\rbrack} \times \left( {\delta_{HV} - \pi} \right)} + {{\frac{1}{2}\left\lbrack {\frac{\pi - \delta_{HV} - \sigma}{{{\pi - \delta_{HV}}} + \sigma} + 1} \right\rbrack} \times \left( {\delta_{HV} + \pi} \right)}}$

However, narrowly defining δ_(RV) to be exactly opposite of δ_(HV) wouldresult in a condition where the two vehicles would almost never beclassified as heading in opposite direction when in reality thiscondition is a very common occurrence. In order to account for smalldifferences in heading angles, the variable φ₂ is used to define a rangea range of heading angles for the RV in which the RV would be consideredto be heading in the opposite direction of the HV. To define this range,the following expressions are defined:

Minimum RV heading angle:

If δ_(RV)−φ₂<0 then δ_(RV) _(min) ¹⁰=2π+δ_(RV)−φ₂

If δ_(RV)−φ₂≥0 then δ_(RV) _(min) ¹⁰=δ_(RV)−φ₂

These conditions can be combined into one mathematical expression as:δ_(RV) _(min) ¹⁰=ζ_(min) ₁ ×(2π−δ_(RV)−φ₂)+ζ_(min) ₁ ×(δ_(RV)−φ₂)where:

$\zeta_{\min_{1}} = {\frac{1}{2}\left\lbrack {\frac{0 - \left( {\delta_{RV} - \varphi_{2}} \right) - \sigma}{{{0 - \left( {\delta_{RV} - \varphi_{2}} \right)}} + \sigma} + 1} \right\rbrack}$$\zeta_{\min_{2}} = {\frac{1}{2}\left\lbrack {\frac{\left( {\delta_{RV} - \varphi_{2}} \right) - 0 + \sigma}{{{\left( {\delta_{RV} - \varphi_{2}} \right) - 0}} + \sigma} + 1} \right\rbrack}$

These expressions have two values, 0 or 1 depending on the value ofδ_(RV) and can be thought of as filtering functions that ensure theappropriate expression is used to calculate the value of δ_(RV) _(min)¹⁰.

Maximum RV heading angle

If δ_(RV)+φ₂<2π then δ_(RV) _(min) ¹⁰=δ_(RV)+φ₂

If δ_(RV)+φ₂≥2π then δ_(RV) _(max) ¹⁰=δ_(RV)+φ₂−2π

These conditions can be combined into one mathematical expression as:δ_(RV) _(max) ¹⁰=ζ_(max) ₁ ×(δ_(RV)+φ₂)+ζ_(max) ₂ ×(δ_(RV)+φ₂−2π

where:

$\zeta_{\max_{1}} = {\frac{1}{2}\left\lbrack {\frac{{2\pi} - \left( {\delta_{RV} + \varphi_{2}} \right) - \sigma}{{{{2\pi} - \left( {\delta_{RV} + \varphi_{2}} \right)}} + \sigma} + 1} \right\rbrack}$$\zeta_{\max_{2}} = {\frac{1}{2}\left\lbrack {\frac{\left( {\delta_{RV} + \varphi_{2}} \right) - {2\pi} + \sigma}{{{\left( {\delta_{RV} + \varphi_{2}} \right) - {2\pi}}} + \sigma} + 1} \right\rbrack}$

These expressions have two values, 0 or 1 depending on the value ofδ_(RV) and can be thought of as filtering functions that ensure theappropriate expression is used to calculate the value of δ_(RV) _(max)¹⁰.

The remote vehicle 14 is considered to be traveling in the directionopposite of the host vehicle 10 when the heading angle of the remotevehicle 14, δ_(RV) falls within the range δ_(RV) _(min) ¹⁰ and δ_(RV)_(max) ¹⁰ therefore cases exist where the heading angle of the hostvehicle 10, δ_(HV) will be less than δ_(RV) _(min) ¹⁰ and less thanδ_(RV) _(min) ¹⁰ when δ_(HV) is less than π as shown in FIGS. 26 and 27where δ_(HV) less than π and less than δ_(RV) _(min) ¹⁰ and δ_(RV)_(max) ¹⁰.

There also exist cases where δ_(HV) will be greater than δ_(RV) _(min)¹⁰ and greater than δ_(RV) _(max) ¹⁰ when δ_(HV) is greater than πotherwise the RV will be considered to be traveling in a direction otherthan the opposite direction of the HV as shown in FIGS. 28 and 29 whereδ_(HV) greater than π and greater than SRS and δ_(RV) _(min) ¹⁰ andδ_(RV) _(max) ¹⁰.

However, because of the fixed reference used where North=0°, there arecases where δ_(HV) will be less than δ_(RV) _(min) ¹⁰ and greater thanδ_(RV) _(max) ¹⁰ when δ_(HV) is less than or greater than π such asshown in FIGS. 30 and 31. FIG. 30, δ_(HV)<π and δ_(RV) _(max)¹⁰<δ_(HV)<δ_(HV)<δ_(RV) _(min) ¹⁰ and in FIG. 31, δ_(HV)>π and δ_(RV)_(max) ¹⁰<δ_(HV)<δ_(RV) _(min) ¹⁰.

Consider the following expressions for H₁ and H₂.

H₁=δ_(HV)−δ_(RV) _(min) ¹⁰

H₂=δ_(HV)−δ_(RV) _(max) ¹⁰

For any value of δ_(HV), the values for H₁ and H₂ fall within threedistinct categories:

1: H₁ is negative, H₂ is negative and H₁>H₂ (δ_(HV)<δ_(RV) _(min) ¹⁰ andδ_(HV)<δ_(RV) _(max) ¹⁰)

2: H₁ is negative, H₂ is positive and H₁<H₂ (δ_(HV)<δ_(RV) _(min) ¹⁰ andδ_(HV)>δ_(RV) _(max) ¹⁰)

3: H₁ is positive, H₂ is positive and H₁>H₂ (δ_(HV)>δ_(RV) _(min) ¹ andδ_(HV)>δ_(RV) _(max) ¹⁰)

From these three conditions, it can be shown that for any combination ofδ_(HV) and δ_(RV), where 0≤δ_(HV)<2π and 0≤δ_(RV)<2π the followingexpressions can be used to identify if the host vehicle 10 and theremote vehicle 14 are traveling in opposite directions.

$\Delta_{1}^{10} = {{\frac{1}{8}\left\lbrack {\frac{\delta_{RV} - \delta_{{RV}_{m\; i\; n}}^{10} + \sigma}{{{\delta_{RV} - \delta_{{RV}_{m\; i\; n}}^{10}}} + \sigma} + 1} \right\rbrack} \times {\quad{{{\left\lbrack {\frac{\delta_{{RV}_{m\; a\; x}}^{10} - \delta_{RV} + \sigma}{{{\delta_{{RV}_{m\; a\; x}}^{10} - \delta_{RV}}} + \sigma} + 1} \right\rbrack \times \left\lbrack {\frac{H_{1} - H_{2} - \sigma}{{{H_{1} - H_{2}}} + \sigma} + 1} \right\rbrack{If}\mspace{14mu} H_{1}} > {H_{2}\mspace{14mu}{and}\mspace{14mu}\delta_{{RV}_{m\; i\; n}}^{10}} \leq \delta_{RV} \leq \delta_{{RV}_{{ma}\; x}}^{10}},{\Delta_{1}^{10} = {{1\mspace{14mu}{otherwise}\mspace{14mu}\Delta_{1}^{10}} = {{0\Delta_{2}^{10}} = {{\frac{1}{8}\left\lbrack {\frac{\delta_{RV} - \delta_{{RV}_{m\; i\; n}}^{10} + \sigma}{{{\delta_{RV} - \delta_{{RV}_{m\; i\; n}}^{10}}} + \sigma} + 1} \right\rbrack} \times {\quad{{{\left\lbrack {\frac{\delta_{RV} - \delta_{{RV}_{m\; a\; x}}^{10} + \sigma}{{{\delta_{RV} - \delta_{{RV}_{m\; a\; x}}^{10}}} + \sigma} + 1} \right\rbrack \times \left\lbrack {1 - \frac{H_{1} - H_{2} - \sigma}{{{H_{1} - H_{2}}} + \sigma}} \right\rbrack{If}\mspace{14mu} H_{1}} < H_{2}},{\delta_{{RV}_{m\; i\; n}}^{10} \leq {\delta_{RV}\mspace{14mu}{and}\mspace{14mu}\delta_{{RV}_{m\;{ax}}}^{10}} \leq \delta_{RV}},{\Delta_{2}^{10} = {{1\mspace{14mu}{otherwise}\mspace{14mu}\Delta_{2}^{10}} = {{0\Delta_{3}^{10}} = {{\frac{1}{8}\left\lbrack {\frac{\delta_{{RV}_{m\; i\; n}}^{10} - \delta_{RV} + \sigma}{{{\delta_{{RV}_{m\; i\; n}}^{10} - \delta_{RV}}} + \sigma} + 1} \right\rbrack} \times {\quad{{{\left\lbrack {\frac{\delta_{{RV}_{m\; a\; x}}^{10} - \delta_{RV} + \sigma}{{{\delta_{{RV}_{m\; a\; x}}^{10} - \delta_{RV}}} + \sigma} + 1} \right\rbrack \times \left\lbrack {1 - \frac{H_{1} - H_{2} - \sigma}{{{H_{1} - H_{2}}} + \sigma}} \right\rbrack{If}\mspace{14mu} H_{1}} < H_{2}}\;,{\delta_{RV} \leq {\delta_{{RV}_{m\; i\; n}}^{10}\mspace{14mu}{and}\mspace{14mu}\delta_{RV}} \leq \delta_{{RV}_{m\;{ax}}}^{10}},{\Delta_{3}^{10} = {{1\mspace{14mu}{otherwise}\mspace{14mu}\Delta_{3}^{10}} = 0}}}}}}}}}}}}}}}}}$

Also, it is advantageous to define the difference of H₁ and H₂ asfollows:H ₁ −H ₂=δ_(HV)−δ_(RV) _(min) ¹⁰−(δ_(HV)−δ_(RV) _(max) ¹⁰)H ₁ −H ₂=δ_(HV)−δ_(RV) _(min) ¹⁰−δ_(HV)+δ_(RV) _(max) ¹⁰H ₁ −H ₂=δ_(HV)−δ_(RV) _(min) ¹⁰−δ_(HV)+δ_(RV) _(max) ¹⁰H ₁ −H ₂=δ_(RV) _(max) ¹⁰−δ_(RV) _(min) ¹⁰

Then the previous expressions can be expressed as:

$\Delta_{1}^{10} = {{\frac{1}{8}\left\lbrack {\frac{\delta_{RV} - \delta_{{RV}_{m\; i\; n}}^{10} + \sigma}{{{\delta_{RV} - \delta_{{RV}_{m\; i\; n}}^{10}}} + \sigma} + 1} \right\rbrack} \times {\quad{{\left\lbrack {\frac{\delta_{{RV}_{m\; a\; x}}^{10} - \delta_{RV} + \sigma}{{{\delta_{{RV}_{m\; a\; x}}^{10} - \delta_{RV}}} + \sigma} + 1} \right\rbrack \times \left\lbrack {\frac{\delta_{{RV}_{m\; a\; x}}^{10} - \delta_{{RV}_{m\; i\; n}}^{10} - \sigma}{{{\delta_{{RV}_{m\; a\; x}}^{10} - \delta_{{RV}_{m\; i\; n}}^{10}}} + \sigma} + 1} \right\rbrack\Delta_{2}^{10}} = {{\frac{1}{8}\left\lbrack {\frac{\delta_{RV} - \delta_{{RV}_{m\; i\; n}}^{10} + \sigma}{{{\delta_{RV} - \delta_{{RV}_{m\; i\; n}}^{10}}} + \sigma} + 1} \right\rbrack} \times {\quad{{\left\lbrack {\frac{\delta_{RV} - \delta_{{RV}_{m\; a\; x}}^{10} + \sigma}{{{\delta_{RV} - \delta_{{RV}_{m\; a\; x}}^{10}}} + \sigma} + 1} \right\rbrack \times \left\lbrack {1 - \frac{\delta_{{RV}_{m\; a\; x}}^{10} - \delta_{{RV}_{m\; i\; n}}^{10} - \sigma}{{{\delta_{{RV}_{m\; a\; x}}^{10} - \delta_{{RV}_{m\; i\; n}}^{10}}} + \sigma}} \right\rbrack\Delta_{3}^{10}} = {{\frac{1}{8}\left\lbrack {\frac{\delta_{{RV}_{m\; i\; n}}^{10} - \delta_{RV} + \sigma}{{{\delta_{{RV}_{m\; i\; n}}^{10} - \delta_{RV}}} + \sigma} + 1} \right\rbrack} \times {\quad{\left\lbrack {\frac{\delta_{{RV}_{m\; a\; x}}^{10} - \delta_{RV} + \sigma}{{{\delta_{{RV}_{m\; a\; x}}^{10} - \delta_{RV}}} + \sigma} + 1} \right\rbrack \times \left\lbrack {1 - \frac{\delta_{{RV}_{m\; a\; x}}^{10} - \delta_{{RV}_{m\; i\; n}}^{10} - \sigma}{{{\delta_{{RV}_{m\; a\; x}}^{10} - \delta_{{RV}_{m\; i\; n}}^{10}}} + \sigma}} \right\rbrack}}}}}}}}}$

By summing these three expressions, it can be determined that the hostvehicle 10 and the remote vehicle 14 are approaching each other fromopposite directions if:

${\sum\limits_{i = 1}^{3}\Delta_{i}^{10}} = {1\left( {{RQ} = 10} \right)}$Thus:

$r_{2} = {\sum\limits_{i = 1}^{3}{\Delta_{i}^{10} \times 1}}$$q_{2} = {\sum\limits_{i = 1}^{3}{\Delta_{i}^{10} \times 0}}$

Host Vehicle and Remote Vehicle approaching from crossing directions(RQ=11)

When the remote vehicle 14 and the host vehicle 10 approach each otherfrom directions that result in a crossing path, the remove vehicleheading angle, δ_(RV) can be defined as a function of host vehicleheading angle, δ_(HV) according to the following expressions. Since acrossing path can occur if the remote vehicle 14 approaches from theleft or right, a total of four angles must be defined; minimum andmaximum angles for the left and minimum and maximum angle for the right.If δ_(RV) falls within the two ranges, a crossing path exists.

Remote Vehicle Heading angle as a function of Host Vehicle heading anglefor the case of vehicles crossing paths can be defined as follows:

Minimum RV heading angle

$\delta_{{RV}_{m\; i\; n\mspace{14mu} L}}^{11} = {{{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - 0 + \sigma}{{{\delta_{HV} - 0}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{\varphi_{6} - \delta_{HV} - \sigma}{{{\varphi_{6} - \delta_{HV}}} + \sigma} + 1} \right\rbrack \times \left( {\delta_{HV} + \varphi_{3}} \right)} + {{\frac{1}{4}\left\lbrack \frac{\delta_{HV} - \varphi_{6} + \sigma}{{{\delta_{HV} - \varphi_{6}}} + \sigma} \right\rbrack} \times \left\lbrack {\frac{{2\pi} - \delta_{HV} - \sigma}{{{{2\pi} - \delta_{HV}}} + \sigma} + 1} \right\rbrack \times \left( {\delta_{HV} - \varphi_{6}} \right)}}$$\delta_{{RV}_{m\; i\; n\mspace{14mu} R}}^{11} = {{{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - 0 + \sigma}{{{\delta_{HV} - 0}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{\varphi_{4} - \delta_{HV} - \sigma}{{{\varphi_{4} - \delta_{HV}}} + \sigma} + 1} \right\rbrack \times \left( {\delta_{HV} + \varphi_{5}} \right)} + {{\frac{1}{4}\left\lbrack \frac{\delta_{HV} - \varphi_{4} + \sigma}{{{\delta_{HV} - \varphi_{4}}} + \sigma} \right\rbrack} \times \left\lbrack {\frac{{2\pi} - \delta_{HV} - \sigma}{{{{2\pi} - \delta_{HV}}} + \sigma} + 1} \right\rbrack \times \left( {\delta_{HV} - \varphi_{4}} \right)}}$

Maximum RV heading angle

$\delta_{{RV}_{m\;{ax}\mspace{14mu} L}}^{11} = {{{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - 0 + \sigma}{{{\delta_{HV} - 0}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{\varphi_{5} - \delta_{HV} - \sigma}{{{\varphi_{5} - \delta_{HV}}} + \sigma} + 1} \right\rbrack \times \left( {\delta_{HV} + \varphi_{4}} \right)} + {{\frac{1}{4}\left\lbrack \frac{\delta_{HV} - \varphi_{5} + \sigma}{{{\delta_{HV} - \varphi_{5}}} + \sigma} \right\rbrack} \times \left\lbrack {\frac{{2\pi} - \delta_{HV} - \sigma}{{{{2\pi} - \delta_{HV}}} + \sigma} + 1} \right\rbrack \times \left( {\delta_{HV} - \varphi_{5}} \right)}}$$\delta_{{RV}_{m\;{ax}\mspace{14mu} R}}^{11} = {{{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - 0 + \sigma}{{{\delta_{HV} - 0}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{\varphi_{3} - \delta_{HV} - \sigma}{{{\varphi_{3} - \delta_{HV}}} + \sigma} + 1} \right\rbrack \times \left( {\delta_{HV} + \varphi_{6}} \right)} + {{\frac{1}{4}\left\lbrack \frac{\delta_{HV} - \varphi_{3} + \sigma}{{{\delta_{HV} - \varphi_{3}}} + \sigma} \right\rbrack} \times \left\lbrack {\frac{{2\pi} - \delta_{HV} - \sigma}{{{{2\pi} - \delta_{HV}}} + \sigma} + 1} \right\rbrack \times \left( {\delta_{HV} - \varphi_{3}} \right)}}$where:

φ₃=π/2−φ_(L)

φ₄=π/2+φ_(L)

φ₅=3π/2−φ_(R)

-   φ₆=3π/2+φ_(R)    -   φ_(L) and φ_(R) are threshold values that defines the angular        range in which the remote vehicle 14 is defined to be in a        crossing path with the host vehicle 10.

These variables define the minimum and maximum boundaries for the rangeof δ_(RV) with respect to δ_(HV) for crossing paths values of δ_(RV)that fall outside these ranges are considered to be another conditionsuch as in-path, opposite path or diverging path. The direction, left orright from which the RV is approaching is immaterial but a singleequation for δ_(RV) _(min) ¹¹ and δ_(RV) _(max) ¹¹ is desired. This canbe achieved by the following two equations:

$\delta_{{RV}_{m\; i\; n}}^{11} = {{\delta_{{RV}_{m\; i\; n\mspace{14mu} L}}^{11} \times {\frac{1}{2}\left\lbrack {\frac{L_{Q_{1}} + L_{Q_{2}} - \sigma}{{{L_{Q_{1}} + L_{Q_{2}}}} + \sigma} + 1} \right\rbrack}} + {\delta_{{RV}_{m\; i\; n\mspace{14mu} R}}^{11} \times {\frac{1}{2}\left\lbrack {\frac{R_{Q_{1}} + R_{Q_{2}} - \sigma}{{{R_{Q_{1}} + R_{Q_{2}}}} + \sigma} + 1} \right\rbrack}}}$$\delta_{{RV}_{{ma}\; x}}^{11} = {{\delta_{{RV}_{{ma}\; x\mspace{14mu} L}}^{11} \times {\frac{1}{2}\left\lbrack {\frac{L_{Q_{1}} + L_{Q_{2}} - \sigma}{{{L_{Q_{1}} + L_{Q_{2}}}} + \sigma} + 1} \right\rbrack}} + {\delta_{{RV}_{{ma}\; x\mspace{14mu} R}}^{11} \times {\frac{1}{2}\left\lbrack {\frac{R_{Q_{1}} + R_{Q_{2}} - \sigma}{{{R_{Q_{1}} + R_{Q_{2}}}} + \sigma} + 1} \right\rbrack}}}$where

$\mspace{20mu}{L_{Q_{1}} = {L_{Q_{4}} = {{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{6} + \sigma}{{{\delta_{HV} - A_{6}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{7} - \delta_{HV} - \sigma}{{{A_{7} - \delta_{HV}}} + \sigma} + 1} \right\rbrack}}}$$L_{Q_{2}} = {L_{Q_{3}} = {{{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - 0 + \sigma}{{{\delta_{HV} - 0}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{13} - \delta_{HV} - \sigma}{{{A_{13} - \delta_{HV}}} + \sigma} + 1} \right\rbrack} + {{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{16} + \sigma}{{{\delta_{HV} - A_{16}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{{2\pi} - \delta_{HV} - \sigma}{{{{2\pi} - \delta_{HV}}} + \sigma} + 1} \right\rbrack}}}$$R_{Q_{1}} = {L_{Q_{4}} = {{{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - 0 + \sigma}{{{\delta_{HV} - 0}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{5} - \delta_{HV} - \sigma}{{{A_{5} - \delta_{HV}}} + \sigma} + 1} \right\rbrack} + {{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{8} + \sigma}{{{\delta_{HV} - A_{8}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{{2\pi} - \delta_{HV} - \sigma}{{{{2\pi} - \delta_{HV}}} + \sigma} + 1} \right\rbrack}}}$$\mspace{20mu}{R_{Q_{2}} = {R_{Q_{3}} = {{\frac{1}{4}\left\lbrack {\frac{\delta_{HV} - A_{14} + \sigma}{{{\delta_{HV} - A_{14}}} + \sigma} + 1} \right\rbrack} \times \left\lbrack {\frac{A_{15} - \delta_{HV} - \sigma}{{{A_{15} - \delta_{HV}}} + \sigma} + 1} \right\rbrack}}}$and:

A₅=β₁−φ₂

A₆=β₁+φ₂

A₇=β₁+π−φ₂

A₈=β₁+π+φ₂

A₁₃=β₁−π−φ₂

A₁₄=β₁−π+φ₂

A₁₅=β₁−φ₂

A₁₆=β₁+φ₂

The remote vehicle 14 is considered to be in a crossing path with thehost vehicle 10 when the heading angle of the remote vehicle 14, δ_(RV)falls within the range δ_(RV) _(min) ¹¹ and δ_(RV) _(max) ¹¹ as definedabove. When the remote vehicle 14 is approaching from the left, thereare three regions that need to be considered:

${0 \leq \delta_{HV} < {\frac{3\pi}{2} - \varphi_{L}}}->\left\{ {{{{\begin{matrix}{\delta_{HV} < \delta_{{RV}_{m\; i\; n}}^{11}} \\{\delta_{HV} < \delta_{{RV}_{m\;{ax}}}^{11}}\end{matrix}\frac{3\pi}{2}} - \varphi_{L}} \leq \delta_{HV} < {\frac{3\pi}{2} + \varphi_{L}}}->\left\{ {{{{\begin{matrix}{\delta_{HV} < \delta_{{RV}_{m\; i\; n}}^{11}} \\{\delta_{HV} > \delta_{{RV}_{{ma}\; x}}^{11}}\end{matrix}\frac{3\pi}{2}} + \varphi_{L}} \leq \delta_{HV} < {2\pi}}->\left\{ \begin{matrix}{\delta_{HV} > \delta_{{RV}_{m\; i\; n}}^{11}} \\{\delta_{HV} > \delta_{{RV}_{{ma}\; x}}^{11}}\end{matrix} \right.} \right.} \right.$

These regions are illustrated in FIGS. 32 and 33, where0≤δ_(HV)<3π/2−φ_(L), in FIGS. 34 and 35 where3π/2−φ_(L)≤δ_(HV)<3π/2+φ_(L), and in FIGS. 36 and 37 where3π/2=φ_(L)≤δ_(HV)<2π.

Similarly, when the remote vehicle 14 is approaching from the right,there are three regions that need to be considered:

${0 \leq \delta_{HV} < {\frac{\pi}{2} - \varphi_{R}}}->\left\{ {{{{\begin{matrix}{\delta_{HV} < \delta_{{RV}_{m\; i\; n}}^{11}} \\{\delta_{HV} < \delta_{{RV}_{m\;{ax}}}^{11}}\end{matrix}\frac{\pi}{2}} - \varphi_{R}} \leq \delta_{HV} < {\frac{\pi}{2} + \varphi_{R}}}->\left\{ {{{{\begin{matrix}{\delta_{HV} < \delta_{{RV}_{m\; i\; n}}^{11}} \\{\delta_{HV} > \delta_{{RV}_{{ma}\; x}}^{11}}\end{matrix}\frac{\pi}{2}} + \varphi_{R}} \leq \delta_{HV} < {2\pi}}->\left\{ \begin{matrix}{\delta_{HV} > \delta_{{RV}_{m\; i\; n}}^{11}} \\{\delta_{HV} > \delta_{{RV}_{{ma}\; x}}^{11}}\end{matrix} \right.} \right.} \right.$

These regions are illustrated in FIGS. 38 and 39 as 0≤δ_(HV)<π/2−φ_(R),in FIGS. 40 and 41 as π/2−φ_(R)≤δ_(HV)<π2+φ_(R), and in FIGS. 42 and 43as π/2+φ_(R)≤δ_(HV)<2π.

Consider the following expressions for H₁ and H₂.H ₁=δ_(HV)−δ_(RV) _(min) ¹¹H ₂=δ_(HV)−δ_(RV) _(max) ¹¹

For any value of δ_(HV), the values for H₁ and H₂ fall within threedistinct categories:

1: H₁ is negative, H₂ is negative and H₁>H₂

2: H₁ is negative, H₂ is positive and H₁<H₂

3: H₁ is positive, H₂ is positive and H₁>H₂

From these three conditions, it can be shown that for any combination ofδ_(HV) and δ_(RV), where 0≤δ_(HV)<2π and 0≤δ_(RV)<2π the followingexpressions can be used to identify if the host vehicle 10 and theremote vehicle 10 are crossing paths.

$\Delta_{1}^{11} = {{\frac{1}{8}\left\lbrack {\frac{\delta_{RV} - \delta_{{RV}_{m\; i\; n}}^{11} + \sigma}{{{\delta_{RV} - \delta_{{RV}_{m\; i\; n}}^{11}}} + \sigma} + 1} \right\rbrack} \times {\quad{{{\left\lbrack {\frac{\delta_{{RV}_{m\; a\; x}}^{11} - \delta_{RV} + \sigma}{{{\delta_{{RV}_{m\; a\; x}}^{11} - \delta_{RV}}} + \sigma} + 1} \right\rbrack \times \left\lbrack {\frac{H_{1} - H_{2} - \sigma}{{{H_{1} - H_{2}}} + \sigma} + 1} \right\rbrack\mspace{20mu}{If}\mspace{14mu} H_{1}} > H_{2}},{\delta_{{RV}_{m\; i\; n}}^{11} \leq \delta_{RV} < \delta_{{RV}_{{ma}\; x}}^{11}},{\Delta_{1}^{11} = {{1\mspace{14mu}{otherwise}\mspace{14mu}\Delta_{1}^{11}} = {{0\Delta_{2}^{11}} = {{\frac{1}{8}\left\lbrack {\frac{\delta_{{RV}_{m\; i\; n}}^{11} - \delta_{RV} + \sigma}{{{\delta_{{RV}_{m\; i\; n}}^{11} - \delta_{RV}}} + \sigma} + 1} \right\rbrack} \times {\quad{{{\left\lbrack {\frac{\delta_{{RV}_{m\; a\; x}}^{11} - \delta_{RV} + \sigma}{{{\delta_{{RV}_{m\; a\; x}}^{11} - \delta_{RV}}} + \sigma} + 1} \right\rbrack \times \left\lbrack {1 - \frac{H_{1} - H_{2} + \sigma}{{{H_{1} - H_{2}}} + \sigma}} \right\rbrack{If}\mspace{14mu} H_{1}} < H_{2}},{\delta_{{RV}_{m\; i\; n}}^{11} \leq {\delta_{RV}\mspace{14mu}{and}\mspace{14mu}\delta_{{RV}_{m\;{ax}}}^{11}} \leq \delta_{RV}},{\Delta_{2}^{11} = {{1\mspace{14mu}{otherwise}\mspace{14mu}\Delta_{2}^{11}} = {{0\Delta_{3}^{11}} = {{\frac{1}{8}\left\lbrack {\frac{\delta_{RV} - \delta_{{RV}_{m\; i\; n}}^{11} + \sigma}{{{\delta_{RV} - \delta_{{RV}_{m\; i\; n}}^{11}}} + \sigma} + 1} \right\rbrack} \times {\quad{{{\left\lbrack {\frac{\delta_{RV} - \delta_{{RV}_{m\; a\; x}}^{11} + \sigma}{{{\delta_{RV} - \delta_{{RV}_{m\; a\; x}}^{11}}} + \sigma} + 1} \right\rbrack \times \left\lbrack {1 - \frac{H_{1} - H_{2} + \sigma}{{{H_{1} - H_{2}}} + \sigma}} \right\rbrack{If}\mspace{14mu} H_{1}} < H_{2}}\;,{\delta_{{RV}_{m\; i\; n}}^{11} \leq {\delta_{RV}\mspace{14mu}{and}\mspace{14mu}\delta_{{RV}_{m\;{ax}}}^{11}} \leq \delta_{RV}},{\Delta_{3}^{11} = {{1\mspace{14mu}{otherwise}\mspace{14mu}\Delta_{3}^{11}} = 0}}}}}}}}}}}}}}}}}$

If H₁<H₂, δ_(RV) _(min) ¹¹≤δ_(RV) and δ_(RV) _(min) ¹¹≤δ_(RV), Δ₃ ¹¹=1otherwise Δ₃ ¹¹=0

Also, it is advantageous to define the difference of H₁ and H₂ asfollows:H ₁ −H ₂δ_(HV)−δ_(RV) _(min) ¹¹−(δH _(V)−δ_(RV) _(max) ¹¹)H ₁ −H ₂δ_(HV)−δ_(RV) _(min) ¹ −δH _(V)−δ_(RV) _(max) ¹¹H ₁ −H ₂δ_(HV)−δ_(RV) _(min) ¹ −δH _(V)−δ_(RV) _(max) ¹¹H ₁ −H ₂=δ_(RV) _(max) ¹¹−δ_(RV) _(min) ¹¹

Then the expressions above can be expressed as:

$\Delta_{1}^{11} = {{\frac{1}{8}\left\lbrack {\frac{\delta_{RV} - \delta_{{RV}_{m\; i\; n}}^{11} + \sigma}{{{\delta_{RV} - \delta_{{RV}_{m\; i\; n}}^{11}}} + \sigma} + 1} \right\rbrack} \times {\quad{{\left\lbrack {\frac{\delta_{{RV}_{m\; a\; x}}^{11} - \delta_{RV} + \sigma}{{{\delta_{{RV}_{m\; a\; x}}^{11} - \delta_{RV}}} + \sigma} + 1} \right\rbrack \times \left\lbrack {\frac{\delta_{{RV}_{m\; a\; x}}^{11} - \delta_{{RV}_{m\; i\; n}}^{11} - \sigma}{{{\delta_{{RV}_{m\; a\; x}}^{11} - \delta_{{RV}_{m\; i\; n}}^{11}}} + \sigma} + 1} \right\rbrack\Delta_{2}^{11}} = {{\frac{1}{8}\left\lbrack {\frac{\delta_{{RV}_{m\; i\; n}}^{11} - \delta_{RV} + \sigma}{{{\delta_{{RV}_{m\; i\; n}}^{11} - \delta_{RV}}} + \sigma} + 1} \right\rbrack} \times {\quad{{\left\lbrack {\frac{\delta_{{RV}_{m\; a\; x}}^{11} - \delta_{RV} + \sigma}{{{\delta_{{RV}_{m\; a\; x}}^{11} - \delta_{RV}}} + \sigma} + 1} \right\rbrack \times \left\lbrack {1 - \frac{\delta_{{RV}_{m\; a\; x}}^{11} - \delta_{{RV}_{m\; i\; n}}^{11} + \sigma}{{{\delta_{{RV}_{m\; a\; x}}^{11} - \delta_{{RV}_{m\; i\; n}}^{11}}} + \sigma}} \right\rbrack\Delta_{3}^{11}} = {{\frac{1}{8}\left\lbrack {\frac{\delta_{RV} - \delta_{{RV}_{m\; i\; n}}^{11} + \sigma}{{{\delta_{RV} - \delta_{{RV}_{m\; i\; n}}^{11}}} + \sigma} + 1} \right\rbrack} \times {\quad{\left\lbrack {\frac{\delta_{RV} - \delta_{{RV}_{m\; a\; x}}^{11} + \sigma}{{{\delta_{RV} - \delta_{{RV}_{m\; a\; x}}^{11}}} + \sigma} + 1} \right\rbrack \times \left\lbrack {1 - \frac{\delta_{{RV}_{m\; a\; x}}^{11} - \delta_{{RV}_{m\; i\; n}}^{11} + \sigma}{{{\delta_{{RV}_{m\; a\; x}}^{11} - \delta_{{RV}_{m\; i\; n}}^{11}}} + \sigma}} \right\rbrack}}}}}}}}}$

By summing these three expressions, it can be determined that the hostvehicle 10 and the remote vehicle 14 are crossing paths if:

${\sum\limits_{i = 1}^{3}\Delta_{i}^{11}} = {1\left( {{RQ} = 11} \right)}$thus:

$r_{3} = {\sum\limits_{i = 1}^{3}{\Delta_{i}^{11} \times 1}}$$q_{3} = {\sum\limits_{i = 1}^{3}{\Delta_{i}^{11} \times 1}}$

and finally:

$R = {\sum\limits_{i = 1}^{3}r_{i}}$$Q = {\sum\limits_{i = 1}^{3}q_{i}}$

If R=Q=0 the paths of the RV and HV are considered to be diverging awayfrom each other.

FIG. 44 identifies the interdependencies of the source data andexpressions that are used to determine the values of the digits Xthrough Q according to the equations discussed above.

Turning back to the flowchart in FIG. 11, in Step 102, the trafficcircle warning system 12 analyzes the relative position code XWVUTSRQobtained by the calculations described above to determine in Step 104whether a warning should be issued. A warning is issued in the followingtwo circumstances.

Host vehicle 10 is approaching traffic circle 40 when the remote vehicle14 is traveling in traffic circle:

Under these conditions, the software application running on thecontroller 22 looks for a relative position code where XWVUTSRQ equals00010011. According to the calculations of this code as discussed above,the remote vehicle 14 is ahead (XW=00) of the host vehicle 10, theremote vehicle 14 is to the left (VU=01) of the host vehicle 10, thehost vehicle 10 and the remote vehicle 14 are at the same elevation(TS=00), and the host vehicle 10 and the remote vehicle 14 are crossingpaths (RQ=11). Thus, this condition represents the condition shown in,for example, FIG. 3. If this condition is true, a threat exists and inStep 106, the controller 22 controls the traffic circle warning system12 to issue a warning according to, for example, the warning logicdescribed with regard to FIG. 45 below. However, if this condition isnot true, the controller 22 controls the traffic circle warning system12 in Step 108 to refrain from issuing a warning.

Host vehicle 10 is traveling in traffic circle 40 as remote vehicle 14approaches the traffic circle 40:

Under these conditions, the host vehicle 10 first ascertains whether thehost vehicle 10 is traveling in the traffic circle 40. The computer 22on the host vehicle 10 can determine this by using information fromsimilarly equipped remote vehicles 10 that have passed through thetraffic circle 40 to determine the existence of the traffic circle 40and calculate the radius of the traffic circle 40. In the absence ofsuch information, the software application operating on controller 22can determine that the host vehicle 10 is traveling in a traffic circle40 according to a similar method used to determine the existence of atraffic circle 40 using information received from remote vehicles 14only in the absence of remote vehicles 14 in the traffic circle 14, andthe host vehicle 10 can use its own GPS position and heading todetermine that the host vehicle 10 is traveling in the traffic circle40.

Once the software application running on the controller 22 in the hostvehicle 10 determines that the host vehicle 10 is in the traffic circle40, the controller 22 looks for a relative position code where XWVUTSRQequals 00100011. According to the calculations of this code as discussedabove, the remote vehicle 14 is ahead (XW=00) of the host vehicle 10,the remote vehicle 14 is to the right (VU=10) of the host vehicle 10,the host vehicle 10 and the remote vehicle 14 are at the same elevation(TS=00), and the host vehicle 10 and the remote vehicle 14 are crossingpaths (RQ=11). Thus, this condition represents the condition shown in,for example, FIG. 7. If this condition is true, a threat exists and inStep 106, the controller 22 controls the traffic circle warning system12 to issue a warning according to, for example, the warning logicdescribed with regard to FIG. 45 below. However, if this condition isnot true, the controller 22 controls the traffic circle warning system12 in Step 108 to refrain from issuing a warning.

As can be appreciated from FIG. 2, such as warning can be a displayedwarning on the screen display 32A, an audio warning via the audiospeaker 32B, a tactile warning, or any other suitable type of warning asunderstood in the art. The traffic circle warning system 12 can alsoprovide an audio indication of the approaching circle via the audiospeaker 32B, a tactile indication, or any other suitable type ofwarning.

FIG. 45 is a flowchart illustrating an example of operations performedby, for example, the controller 22 to control the traffic circle warningsystem 12 to issue a warning. For purposes of this flowchart the hostvehicle 10 is referred to as the subject vehicle SV, and the remotevehicle 14 is referred to at the threat vehicle TV. In Step 200, thecontroller 22 determines whether a traffic circle 40 exists in anysuitable manner as discussed herein. For example, the controller candetermine whether a traffic circle 40 exists according to the processesdescribed in U.S. patent application Ser. No. 15/477,827, entitled“Traffic Circle Identification System and Method,” referenced above. Ifthe controller 22 determines that a traffic circle 40 does not exist,the processing continues to Step 202, where the controller 22 determineswhether the remote vehicle 14 is making a left turn across path from theopposite direction (LTAP/OD). If the remote vehicle 14 is not makingsuch a left turn, the processing continues to Step 204 when thecontroller 22 determines whether the velocity V_(SV) of the host vehicle10 (e.g., in meters per second) is below a suitable thresholdV_(threshold), which is speed threshold value, in meters per second(e.g., 2 meters per second), related to the host vehicle speed.Naturally, the threshold value can be V_(threshold) any suitable speedas understood in the art.

If the controller 22 determines in Step 204 that the velocity V_(SV) ofthe host vehicle 10 is not below the threshold V_(threshold), thecontroller 22 determines in Step 206 whether the value of ΔTTC is lessthan the time threshold value of ΔTTC_(T). For purposes of thedescription herein, the value of ΔTTC represents the difference, inseconds, between TTC_(SV) and TTC_(TV). TTC_(SV) represents the time, inseconds, the host vehicle 10 is from the calculated point ofintersection of the paths of the host vehicle 10 and the remote (ortarget) vehicle 14. TTC_(TV) represents the time, in seconds, that theremote vehicle 14 is from the calculated point of intersection of thepaths of the host vehicle 10 and the remote (or target) vehicle 14.ΔTTC_(T) represents a time threshold value, in seconds (e.g., 2 secondsor any suitable value), related to the difference between TTC_(SV) andTTC_(TV).

If the controller 22 determines in Step 206 that value of ΔTTC is notless than the time threshold value of ΔTTC_(T), the processing returnsto the beginning. However, if the value of ΔTTC is less than the timethreshold value of ΔTTC_(T), the controller 22 determines in Step 208whether the value of TTC_(SV) is less than a value for TTC_(SVwarn).TTC_(SVwarn) represents a time threshold value, in seconds (e.g., 3seconds or any suitable value), related to how many seconds that thehost vehicle 10 is from the calculated point of intersection of thepaths of the host vehicle 10 and the remote (or target) vehicle 14.Thus, TTC_(SVwarn) defines when a warning should be issued to the driverof the host vehicle 10, and is applicable when both host vehicle 10 andthe remote vehicle 14 are in motion. If the processor determines in Step208 that the value of TTC_(SV) is not less than a value forTTC_(SVwarn), the processing determines in Step 210 whether value ofTTC_(SV) is less than a value for TTC_(SVinform). TTC_(SVinform)represents a time threshold value, in seconds (e.g., 6 seconds or anysuitable value), related to how many seconds that the host vehicle 10 isfrom the calculated point of intersection of the paths of the hostvehicle 10 and the remote (or target) vehicle 14. TTC_(SVinform) defineswhen an informative advisory should be issued to the driver of the hostvehicle 10, and is applicable when both host vehicle 10 and the remotevehicle 14 are in motion. If the controller 22 determines in Step 210that value of TTC_(SV) is not less than a value for TTC_(SVinform), theprocessing returns to the beginning.

However, if the controller 22 determines in Step 208 that the value ofTTC_(SV) is less than a value for TTC_(SVwarn), the controller 22determines in Step 212 whether the host vehicle 10 has activated itsbrakes. If not, the controller 22 determines in Step 214 whether theremote vehicle 14 has activated its brakes based on, for example, thereceived remote vehicle information. If not, the controller 22determines in Step 216 whether the informing by the controller 22 isactive. If not, the controller 22 determines in Step 218 whether thewarning is active. If not, the controller 22 controls the traffic circlewarning system 12 to issue a warning as discussed above in Step 220. Theprocessing then returns to the beginning. However, if the controller 22determines in Step 218 that the warning is active, the processingreturns to the beginning. Also, if the controller 22 determines in Step216 that the informing is active, the controller 22 resets the informingin Step 222, issues the warning in Step 220, and returns to thebeginning.

Looking back at Step 214, if the controller 22 determines that theremote vehicle 14 has activated its brakes, the controller calculatesthe remote vehicle braking in Step 224, namely, the value TVl_(braking)which represents the stopping distance, in meters, for the remotevehicle 14. The controller 22 determines in Step 226 whether the valueTVl_(braking) is less than l_(TV) which represents the distance, inmeters, between the remote vehicle 14 and the point of intersectionbetween the paths of the host vehicle 10 and the remote vehicle 14. Ifthe value TVl_(braking) is less than l_(TV), the controller 22determines in Step 228 whether the informing is active. If so, theprocessing returns to the beginning. However, if the informing is notactive, the controller 22 controls the traffic circle warning system 10to inform the driver of the remote vehicle 14 in Step 230, and theprocessing returns to the beginning. However, if the controller 22determines in Step 226 that the value TVl_(braking) is not less thanl_(TV), the controller 22 processing continues to Step 216 and proceedsas discussed above.

Looking back at Step 212, if the controller 22 determines that the hostvehicle 10 has activated its brakes, the processing continues to Step232 where the controller 22 calculates SVl_(braking) which representsthe stopping distance, in meters, for the host vehicle 10. Thecontroller 22 then determines in Step 234 whether SVl_(braking) is lessthan l_(SV) which represents the distance, in meters, between the hostvehicle 10 and the point of intersection between the paths of the hostvehicle 10 and the remote vehicle 14. If SVl_(braking) is not less thanl_(SV), the processing continues to Step 216 and proceeds as discussedabove. However, if SVl_(braking) is less than l_(SV), the processingcontinues to Step 228 and proceeds as discussed above.

Looking back at Step 210, if the controller determines in Step 210 thatTTC_(SV) is less than a value for TTC_(SVinform), the processingcontinues to Step 226 and proceeds as discussed above.

Looking back at Step 204, if the controller 22 determines that thevelocity V_(SV) of the host vehicle 10 is below the thresholdV_(threshold), the processing continues to Step 238 where the controller22 determines whether the value of TTC_(TV) is less than a value forTTC_(TVwarn). TTC_(TVwarn) represents a time threshold value, in seconds(e.g., 3 seconds or any suitable value), related to how many seconds theremote vehicle 14 (or target vehicle 10) is from the calculated point ofintersection of the paths of the host vehicle 10 and the remote vehicle14. TTC_(TVwarn) defines when a warning should be issued to the driverof the host vehicle 10, and is applicable when host vehicle 10 isstationary and remote vehicle 14 is in motion.

If the controller 22 determines in Step 238 that the value of TTC_(TV)is not less than the value for TTC_(TVwarn), the controller 22determines in Step 236 whether TTC_(TV) of the remote vehicle 14 is lessthan a value for TTC_(TVinform). TTC_(TVinform) represents a timethreshold value, in seconds (e.g., 6 seconds or any suitable value),related to how many seconds the host vehicle 10 is from the calculatedpoint of intersection of the paths of the host vehicle 10 and the remote(or target) vehicle 14. TTC_(TVinform) defines when an informativeadvisory should be issued to the driver of the host vehicle 10, and isapplicable when host vehicle 10 is stationary and the remote vehicle 14is in motion. If TTC_(TV) is not less than a value for TTC_(TVinform),the processing returns to the beginning. However, if TTC_(T)v is lessthan a value for TTC_(TVinform), the processing continues to Step 228and proceeds as discussed above.

However, if the controller 22 determines in Step 238 that the value ofTTC_(TV) is less than the value for TTC_(TVwarn), the processingcontinues to Step 240 where the controller 22 determines whether l_(SV)is less than a suitable value, which in this example is 35 m. If not,the processing continues to Step 236 and proceeds as discussed above.However, if the value is less, the processing continues to Step 242where the controller 22 determines if the brake of the remote vehicle 14is released. If the brake is not released, the processing returns to thebeginning.

However, if the brake is released, the processing continues to Step 244where the controller 22 determines whether the informing in active. Ifthe informing is active, the controller 22 resets the informing in Step246 and continues to Step 248. If the informing is not active, thecontroller 22 continues to Step 248. In Step 248, the controller 22determines whether the warning is active. If the warning is not active,the controller 22 controls the traffic circle warning system 12 to issuethe warning in Step 250 as discussed above, and proceeds to Step 252.However, if the warning is active, the processing proceeds to Step 252.In Step 252, the controller 22 determines whether the brake of the hostvehicle 10 is applied. If not, the processing returns to the beginning.However, if the brake is applied, the controller 22 resets the warningin Step 254 and the processing returns to the beginning.

Looking back at Step 202, if the controller determines that the remotevehicle 14 is making a left turn across path from the oppositedirection, the controller determines in Step 256 whether a value of TTC′is less than a value of TTC_(LTAP2). TTC′ represents a time threshold,in seconds, between the host vehicle 10 and the remote vehicle 14 whenthe remote vehicle 14 is approaching the host vehicle 10 from theopposite direction. TTC_(LTAP2) represents a time threshold, value inseconds (e.g., 3 sec or any suitable value), related to how many secondsthat the remote (or target) vehicle 14 is from the plane perpendicularto the front of the host vehicle 10. TTC_(LTAP2) defines when a warningshould be issued to the driver of the host vehicle 10.

If the controller determines in Step 256 that TTC′ is not less than avalue of TTC_(LTAP2), the processing continues to Step 258 where thecontroller 22 determines whether a value of TTC′ is less than a value ofTTC_(LTAP1). TTC_(LTAP1) a time threshold value, in seconds (e.g., 6seconds or any suitable value), related to how many seconds that theremote (or target) vehicle 14 is from the plane perpendicular to thefront of the host vehicle 10. TTC_(LTAP1) defines when an informativeadvisory should be issued to the driver of the host vehicle 10. If thecontroller determines in Step 258 that TTC′ is not less than a value ofTTC_(LTAP1), the processing returns to the beginning. However, if thevalue is less, the processing continues to Step 228 and proceeds asdiscussed above.

If the controller 22 determines in Step 256 that TTC′ is less than thevalue of TTC_(LTAP2), the processing continues to Step 260 where thecontroller 22 determines whether the velocity V_(SV) of the host vehicle10 is below the threshold V_(threshold). If the value is less, theprocessing continues to Step 242 and proceeds as discussed above.However, if the value is not less, the controller 22 calculates a valuefor a warning variable W in Step 262, and determines if a value for awarning variable W is equal to 1 in Step 264. If the value is not equalto 1, the processing continues to Step 228 and proceeds as discussedabove. However, if the value is equal to 1, the processing continues toStep 216 and proceeds as discussed above.

Looking back at Step 200, if the controller 22 determines that thetraffic circle 40 exists, the controller 22 determines in Step 266whether the RP code XWVUTSRQ is the decimal value 19, which correspondsto the binary value 00010011 as discussed above. If so, the processingcontinues to Step 204 and proceeds as discussed above. However, if thevalue of the RP code is not decimal value 19, the controller 22determines in Step 268 whether the RP code XWVUTSRQ is the decimal value35, which corresponds to the binary value 00100011 as discussed above.If so, the processing continues to Step 204 and proceeds as discussedabove. However, if the value of the RP code is not decimal value 35,then the processing returns to the beginning, and the controller 22controls the traffic circle warning system 12 to refrain from issuing awarning.

While only selected embodiments have been chosen to illustrate thepresent invention, it will be apparent to those skilled in the art fromthis disclosure that various changes and modifications can be madeherein without departing from the scope of the invention as defined inthe appended claims. The functions of one element can be performed bytwo, and vice versa. The structures and functions of one embodiment canbe adopted in another embodiment. It is not necessary for all advantagesto be present in a particular embodiment at the same time. Every featurewhich is unique from the prior art, alone or in combination with otherfeatures, also should be considered a separate description of furtherinventions by the applicant, including the structural and/or functionalconcepts embodied by such feature(s). Thus, the foregoing descriptionsof the embodiments according to the present invention are provided forillustration only, and not for the purpose of limiting the invention asdefined by the appended claims and their equivalents.

What is claimed is:
 1. A traffic circle warning system comprising: anelectronic controller configured to determine whether a traffic circleexists along a current travel path of the host vehicle based on remotevehicle information representing a travel condition of at least oneremote vehicle and, upon determining that the traffic circle exists,evaluate a travel condition of the host vehicle relative to the trafficcircle and the travel condition of the remote vehicle to determinewhether to control a warning system onboard the host vehicle to issue awarning, the travel condition of the at least one remote vehicleindicating an arcuate path of the at least one remote vehicle.
 2. Thetraffic circle warning system according to claim 1, wherein theelectronic controller is configured to control the warning system toissue the warning upon determining based on the travel condition of thehost vehicle and the travel condition of the remote vehicle that adistance between the host vehicle the remote vehicle is decreasing. 3.The traffic circle warning system according to claim 1, wherein theelectronic controller is configured to control the warning system toissue the warning upon determining based on the travel condition of thehost vehicle and the travel condition of the remote vehicle that atravel path of the host vehicle and a travel path of the remote vehicleintersect each other within the traffic circle.
 4. The traffic circlewarning system according to claim 1, wherein the electronic controlleris configured to control the warning system to issue the warning upondetermining based on the travel condition of the host vehicle and thetravel condition of the remote vehicle that the host vehicle isapproaching a traffic entry location of the traffic circle and theremote vehicle is travelling within the traffic circle at apredetermined distance from the traffic entry location.
 5. The trafficcircle warning system according to claim 1, wherein the electroniccontroller is configured to control the warning system to issue thewarning upon determining based on the travel condition of the hostvehicle and the travel condition of the remote vehicle that the remotevehicle is approaching a traffic entry location of the traffic circleand the host vehicle is travelling within the traffic circle at apredetermined distance from the traffic entry location.
 6. The trafficcircle warning system according to claim 1, wherein the electroniccontroller is configured to identify sections of the traffic circle, andto control the warning system to issue the warning upon determiningbased on the travel condition of the host vehicle and the travelcondition of the remote vehicle that the host vehicle is approaching oneof the sections of the traffic circle and the remote vehicle istravelling within the one of the sections of the traffic circle.
 7. Thetraffic circle warning system according to claim 1, wherein theelectronic controller is configured to identify sections of the trafficcircle, and to control the warning system to issue the warning upondetermining based on the travel condition of the host vehicle and thetravel condition of the remote vehicle that the remote vehicle isapproaching one of the sections of the traffic circle and the hostvehicle is travelling within the one of the sections of the trafficcircle.
 8. The traffic circle warning system according to claim 1,wherein the remote vehicle information includes information representinga heading of a remote vehicle.
 9. The traffic circle warning systemaccording to claim 8, wherein the remote vehicle information includesinformation representing a turning radius of the remote vehicle.
 10. Thetraffic circle warning system according to claim 1, wherein theelectronic controller is configured to evaluate a travel condition ofthe host vehicle relative to the traffic circle and the travel conditionof the remote vehicle based on whether the remote vehicle is ahead ofthe host vehicle, whether the remote vehicle is to the left of the hostvehicle, and whether the host vehicle and the remote vehicle are at thesame elevation.
 11. The traffic circle warning system according to claim1, wherein the electronic controller is configured to establishcoordinate areas about the host vehicle, and evaluates the travelcondition of the remote vehicle within a respective one of thecoordinate areas.
 12. The traffic circle warning system according toclaim 1, wherein the electronic controller is configured to determine alocation of the traffic circle relative to the location of the hostvehicle at a predetermined time when the electronic controllerdetermines that the traffic circle exists.
 13. The traffic circlewarning system according to claim 12, wherein the electronic controlleris configured to determine the location of the traffic circle relativeto the location of the host vehicle and a location of the remote vehicleat the predetermined time when the electronic controller determines thatthe traffic circle exists.
 14. A traffic circle warning systemcomprising: an electronic controller configured to determine whether atraffic circle exists along a current travel path of the host vehiclebased on remote vehicle information representing a travel condition ofat least one remote vehicle and, upon determining that the trafficcircle exists, evaluate a travel condition of the host vehicle relativeto the traffic circle and the travel condition of the remote vehicle todetermine whether to control a warning system onboard the host vehicleto issue a warning, the receiver being configured to receive the remotevehicle information via direct communication with the at least oneremote vehicle.
 15. A traffic circle warning method comprising:determining, by an electronic controller, whether a traffic circleexists along a current travel path of the host vehicle based on remotevehicle information representing a travel condition of at least oneremote vehicle, the travel condition of the at least one remote vehicleindicating an arcuate path of the at least one remote vehicle; and upondetermining that the traffic circle exists, evaluating by the electroniccontroller a travel condition of the host vehicle relative to thetraffic circle and the travel condition of the remote vehicle todetermine whether to control a warning system onboard the host vehicleto issue a warning.
 16. The method according to claim 15, furthercomprising controlling, by the electronic controller, the warning systemto issue the warning upon determining based on the travel condition ofthe host vehicle and the travel condition of the remote vehicle that adistance between the host vehicle the remote vehicle is decreasing. 17.The method according to claim 15, further comprising controlling, by theelectronic controller, the warning system to issue the warning upondetermining based on the travel condition of the host vehicle and thetravel condition of the remote vehicle that a travel path of the hostvehicle and a travel path of the remote vehicle intersect each otherwithin the traffic circle.
 18. The method according to claim 15, furthercomprising controlling, by the electronic controller, the warning systemto issue the warning upon determining based on the travel condition ofthe host vehicle and the travel condition of the remote vehicle that thehost vehicle is approaching a traffic entry location of the trafficcircle and the remote vehicle is travelling within the traffic circle ata predetermined distance from the traffic entry location.
 19. The methodaccording to claim 15, further comprising controlling, by the electroniccontroller, the warning system to issue the warning upon determiningbased on the travel condition of the host vehicle and the travelcondition of the remote vehicle that the remote vehicle is approaching atraffic entry location of the traffic circle and the host vehicle istravelling within the traffic circle at a predetermined distance fromthe traffic entry location.
 20. The method according to claim 15,wherein the remote vehicle information includes information representingat least one of a heading of a remote vehicle and a turning radius ofthe remote vehicle.